From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics : Volume 1

Volume 1 The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subjec...

Author: Diogo Arsenio | Laure Saint-Raymond

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Volume 1 The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subject to self-induced electromagnetic forces. At the macroscopic scale, in the incompressible viscous fluid limit the evolution of the plasma is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing that may take on various forms depending on the number of species and on the strength of the interactions. From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way. The first part of this work is devoted to the systematic formal analysis of viscous hydro dynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature. In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman– Enskog-type decomposition and regularity in the v variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more). The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known.

ISBN 978-3-03719-193-4

www.ems-ph.org

Monographs / Arsénio/Saint-Raymond Vol. 1 | Font: Rotis Semi Sans | Farben: Pantone 116, Pantone 287 | RB 24 mm

Diogo Arsénio and Laure Saint-Raymond

From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics

From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics. Volume 1

Diogo Arsénio Laure Saint-Raymond

Monographs in Mathematics


From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics Volume 1

EMS Monographs in Mathematics Edited by Hugo Duminil-Copin (Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France) Gerard van der Geer (University of Amsterdam, The Netherlands) Thomas Kappeler (University of Zürich, Switzerland) Paul Seidel (Massachusetts Institute of Technology, Cambridge, USA)

EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and selfcontained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature.

Previously published in this series: Richard Arratia, A.D. Barbour and Simon Tavaré, Logarithmic Combinatorial Structures: A Probabilistic Approach Demetrios Christodoulou, The Formation of Shocks in 3-Dimensional Fluids Sergei Buyalo and Viktor Schroeder, Elements of Asymptotic Geometry Demetrios Christodoulou, The Formation of Black Holes in General Relativity Joachim Krieger and Wilhelm Schlag, Concentration Compactness for Critical Wave Maps Jean-Pierre Bourguignon, Oussama Hijazi, Jean-Louis Milhorat, Andrei Moroianu and Sergiu Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry Kazuhiro Fujiwara and Fumiharu Kato, Foundations of Rigid Geometry I Demetrios Christodoulou, The Shock Development Problem



Authors: Diogo Arsénio Institut de Mathématiques de Jussieu – Paris Rive-Gauche CNRS & Université Paris Diderot Bâtiment Sophie Germain 8 place Aurélie Nemours 75205 Paris CEDEX 13 France

Laure Saint-Raymond Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon CEDEX 07 France E-mail: [email protected]

E-mail: [email protected]

2010 Mathematics Subject Classification: 76P05, 76W05, 82C40, 35B25 Key words: Plasma, magneto-hydro-dynamics, fluid limits, kinetic theory, entropy method, moment method, hypoellipticity, electromagetic waves, Ohm’s law

ISBN 978-3-03719-193-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2019 European Mathematical Society

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European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich, Switzerland

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Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface The present book aims at presenting in a systematic, painstaking and rather exhaustive manner the incompressible viscous fluid limits of the system of Vlasov–Maxwell– Boltzmann equations for one or two species. In these regimes, the evolution of the fluid is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing. Depending on the precise scaling, this forcing term takes on various forms: it may be linear or nonlinear, electrostatic or governed by some hyperbolic wave equation, possibly constrained by some relation of Ohm’s type. From the mathematical point of view, these models have very different behaviors; in particular, to establish the existence and stability of solutions require sometimes to work with very weak notions of solutions. The asymptotic analysis, which consists most often in retrieving the structure of the limiting system in the scaled Vlasov– Maxwell–Boltzmann system, uses therefore various mathematical methods with important technical refinements. Thus, in order to make the reading easier, different tools will be presented in separate chapters. The first part of this work is devoted to the systematic formal analysis of viscous hydrodynamic limits. Chapter 1 introduces the Vlasov–Maxwell–Boltzmann system as well as its formal properties. An important point to be noted is that the a priori bounds coming from these physical laws do not suffice for proving the existence of global solutions, even in the renormalized sense of DiPerna and Lions [30], which is a major difficulty for the study of fast relaxation limits. This actually explains the dividing of the three other parts of this book, of increasing difficulty, giving rigorous convergence results in more and more general settings. Chapter 2 introduces the different scaling parameters arising in the system, and details the formal steps leading to the constraint relations and the evolution equations in each regime. We thus obtain a rather precise classification of physically relevant models for viscous incompressible plasmas, some of which actually do not seem to have been previously described in the literature. Chapter 3 presents a mathematical analysis of these different models. The most singular of them have a behavior that is actually more similar to the incompressible Euler equations than to the Navier–Stokes equations: the lack of weak stability does not allow to prove the existence of global solutions, with the exception of very weak solutions in the spirit of the dissipative solutions introduced by Lions for the Euler equations [59]. This lack of stability for limiting systems is the second major difficulty encountered in the study of hydrodynamic limits. The goal of the second part is to make precise and rigorous the convergence results described formally in the first part. In order to isolate the difficulties which are specific to the asymptotic analysis, we choose here to prove first conditional results, i.e., to consider the convergence of renormalized solutions even though their existence is not known. This of course does not imply the convergence of weaker solutions, which

vi will be studied in the sequel (renormalized solutions with defect measure, and a fortiori solutions with Young measures), but most of the proof will remain unchanged. The important point is that the analysis is based essentially on the uniform estimates coming from the scaled entropy inequality, which holds in all situations. Furthermore, we will focus exclusively on two typical regimes, namely, leading: from the one-species Vlasov–Maxwell–Boltzmann equations to the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (Theorem 4.5); from the two-species Vlasov–Maxwell–Boltzmann equations to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law in the case of strong interspecies collisions (Theorem 4.7), or to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law in the case of weak interspecies collisions (Theorem 4.6). These asymptotic regimes are critical, in the sense that they are the most singular ones among the formal asymptotics mentioned in Chapter 2 and that all remaining regimes can be treated rigorously by similar or even simpler arguments. We will not detail in this preface the content of all chapters of the second part, but rather insist on the main points requiring a treatment different from the usual hydrodynamic limits [70]. In the case with only one species, the main difference is due to the fact that the transport equation contains force terms involving a derivative with respect to v, which does not allow to transfer equi-integrability from the vvariable to the x-variable as in [37]. This is a major complication. The new idea here consists in getting first some strong compactness in v by using regularizing properties of the gain operator [53] and, then, in transferring this strong compactness to the spatial variable by means of refined hypoelliptic arguments developed in [7]. The second important difference comes from the fast temporal oscillations which couple acoustic and electromagnetic modes. Note that we introduce here a simple method to avoid dealing with non-local projections. Overall, we are eventually able to establish through weak compactness methods a very general result (Theorem 4.5) asserting the convergence of renormalized solutions of the one-species Vlasov–Maxwell–Boltzmann system towards weak solutions of corresponding macroscopic systems. In the case of two species, the situation not only requires to exploit the methods for one species, it is considerably more complex: First of all, there is an additional scaling parameter measuring the strength of interspecies interactions (and, incidentally, the typical size of the electric current, which can be much smaller than the bulk velocities of each of the two species of particles): this implies that the (formal) expansions involve a larger number of terms (for instance, the constraint equations are derived at different orders). Secondly, the linearized collision operator has a more complicated vectorial structure. The inversion of fluxes and the computation of dissipation terms in the limiting energy inequalities are therefore substantially more technical.

vii In the most singular regimes, we get nonlinear constraint equations. This means that renormalization methods, compensated compactness techniques and controls on the conservation defects are already required at this stage of the proof. We have no sufficient uniform a priori bound on the electric current to handle nonlinear terms, which prevents us from taking limits in the approximate conservation of momentum law. To avoid this difficulty, we need to introduce a modified conservation law involving the Poynting vector. Even in this more suitable form, the evolution equations are not stable under weak convergence, and we have no equi-integrability in these singular regimes. We therefore develop an improved modulated entropy method, which allows to consider renormalized solutions without important restrictions on the initial data. Note that this renormalized modulated entropy method should also lead to some improvements concerning the convergence of the Boltzmann equation (without any electromagnetic field) to the Navier–Stokes equations for ill-prepared initial data. The third and fourth parts (which will be published in a second volume) are more technical. They show how to adapt the arguments presented in the conditional case of the second part to take into account the state of the art Cauchy theory for the Vlasov– Maxwell–Boltzmann system. In the case of long-range microscopic interactions giving rise to a collision crosssection with a singularity for grazing collisions, treated in the third part, we start by proving the existence of renormalized solutions with a defect measure in the spirit of the construction by Alexandre and Villani [1]. This result, which is important independently of the study of hydrodynamic limits, has been addressed in the note [8]. The study of hydrodynamic limits follows then essentially the lines of [4] (combined with the results of the conditional part). We would like however to mention some important novelties: The first one concerns the estimate of the defect measure. A refined analysis of the convergence of approximate solutions to the Vlasov–Maxwell– Boltzmann system shows that the defect measure can be controlled by the entropy dissipation. This remark allows for a simplification of the proofs from [4], especially the passage to the limit in the kinetic equation leading to the characterization of the limiting form of the dissipation, and the control of conservation defects. The other simplification is related to the renormalization process. Here we choose a decomposition of the renormalized collision operator which allows both to control the singularity due to the collision cross-section, and to preserve the good scalings for the fluctuation. In particular, the same decomposition can be used for the control of the transport and of the conservation defects (with a loop estimate).

viii In the case of general microscopic interactions (including for instance the case of hard spheres), it is not known how to prove the convergence of approximation schemes of the Vlasov–Maxwell–Boltzmann system, due to a lack of compactness produced by the electromagnetic interaction. The existence of renormalized solutions is therefore still an open problem. Nevertheless, Lions [55] has defined a very weak notion of solution – the measure-valued renormalized solution – defined as limit of approximate solutions: the equation to be satisfied involves indeed Young measures. In the fourth part, we begin by refining the control of Young measures for such solutions by the entropy inequality. We then proceed by showing that the estimates obtained in the second part are very stable, so that they can be generalized with Young measures. By using convexity properties and Jensen inequalities, we can extend all the arguments, and operate both the moment method and the entropy method in more singular regimes. This extension to solutions of the Vlasov–Maxwell–Boltzmann system defined in a very weak sense shows that the methods based on the entropy inequality are extremely robust, and that the convergence is essentially determined by the limiting system. These good asymptotic properties seem to further indicate that the measure-valued solutions defined by Lions (which have never been really studied from the qualitative point of view) are relevant in some sense.

Paris, France, January 2016

Diogo Arsénio & Laure Saint-Raymond

Contents I Formal derivations and macroscopic weak stability . . . . . . . 1 The Vlasov–Maxwell–Boltzmann system 1.1 The Boltzmann collision operator . . . 1.2 Formal macroscopic properties . . . . 1.3 The mathematical framework . . . . .

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2 Scalings and formal limits . . . . . . . . . . . . . . . . . . . . . 2.1 Incompressible viscous regimes . . . . . . . . . . . . . . . . 2.2 Scalings for the electromagnetic field . . . . . . . . . . . . . 2.3 Formal analysis of the one-species asymptotics . . . . . . . 2.3.1 Thermodynamic equilibrium . . . . . . . . . . . . . 2.3.2 Macroscopic constraints . . . . . . . . . . . . . . . . 2.3.3 Evolution equations . . . . . . . . . . . . . . . . . . 2.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 The Vlasov–Poisson–Boltzmann system . . . . . . . 2.4 Formal analysis of the two-species asymptotics . . . . . . . 2.4.1 Thermodynamic equilibrium . . . . . . . . . . . . . 2.4.2 The case of very weak interspecies collisions . . . . . 2.4.3 Macroscopic hydrodynamic constraints . . . . . . . . 2.4.4 Hydrodynamic evolution equations . . . . . . . . . . 2.4.5 Macroscopic electrodynamic constraints and evolution 2.4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 The two-species Vlasov–Poisson–Boltzmann system .

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15 15 17 21 22 25 27 33 36 37 39 46 55 56 60 65 69

3 Weak stability of the limiting macroscopic systems . . . . . . . . . . . . 73 3.1 The incompressible quasi-static Navier–Stokes–Fourier–Maxwell– Poisson system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 The two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with (solenoidal) Ohm’s law . . . . . . . . . . . . . . . . . . 77 3.2.1 Large global solutions in two dimensions . . . . . . . . . . . . 81 3.2.2 Small global solutions in three dimensions . . . . . . . . . . . 83 3.2.3 Weak-strong stability and dissipative solutions . . . . . . . . . 84 3.2.3.1 The incompressible Navier–Stokes–Maxwell system 85 3.2.3.2 The two-fluid incompressible Navier–Stokes– Maxwell system with Ohm’s law . . . . . . . . . . 92 3.2.3.3 The two-fluid incompressible Navier–Stokes– Maxwell system with solenoidal Ohm’s law . . . . 105

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II Conditional convergence results . . . . . . . . . . . . . . . . . . . 119 4

Two typical regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Renormalized solutions . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Vlasov–Boltzmann equation . . . . . . . . . . . . . . . 4.1.2 Coupling the Boltzmann equation with Maxwell’s equations . 4.1.3 The setting of our conditional study . . . . . . . . . . . . . . 4.1.4 Macroscopic conservation laws . . . . . . . . . . . . . . . . 4.2 The incompressible quasi-static Navier–Stokes–Fourier–Maxwell– Poisson system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with (solenoidal) Ohm’s law . . . . . . . . . . . . . . . . . 4.3.1 Weak interactions . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Strong interactions . . . . . . . . . . . . . . . . . . . . . . . 4.4 Outline of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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121 122 122 131 132 134

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5

Weak compactness and relaxation estimates . . . . 5.1 Controls from the relative entropy bound . . . . 5.2 Controls from the entropy dissipation bound . . 5.3 Relaxation towards thermodynamic equilibrium 5.3.1 Infinitesimal Maxwellians . . . . . . . . 5.3.2 Bulk velocity and temperature . . . . . . 5.4 Improved integrability in velocity . . . . . . . .

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157 158 161 164 168 170 174

6

Lower-order linear constraint equations and energy inequalities . . . 6.1 Macroscopic constraint equations for one species . . . . . . . . . . 6.2 Macroscopic constraint equations for two species, weak interactions 6.3 Energy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The limiting Maxwell equations . . . . . . . . . . . . . . . . . . .

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183 183 186 190 196

7

Strong compactness and hypoellipticity . . . . . . . . . . . . . . . . . 7.1 Compactness with respect to v . . . . . . . . . . . . . . . . . . . . 7.1.1 Compactness of the gain term . . . . . . . . . . . . . . . . . 7.1.2 Relative entropy, entropy dissipation and strong compactness 7.2 Compactness with respect to x . . . . . . . . . . . . . . . . . . . . 7.2.1 Hypoellipticity and the transfer of compactness . . . . . . . 7.2.2 Compactness of fluctuations for one species . . . . . . . . . 7.2.3 Compactness of fluctuations for two species . . . . . . . . .

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199 200 200 203 207 208 213 221

8

Higher-order and nonlinear constraint equations . . . . . . . . . . . . 8.1 Macroscopic constraint equations for two species, weak interactions 8.1.1 Proof of Proposition 8.1 . . . . . . . . . . . . . . . . . . . . 8.1.1.1 An admissible renormalization . . . . . . . . . . 8.1.1.2 Convergence of conservation defects . . . . . . . 8.1.1.3 Decomposition of flux terms . . . . . . . . . . .

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233 233 235 235 237 242

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8.1.1.4 Decomposition of acceleration terms . . . . . . . 8.1.1.5 Convergence . . . . . . . . . . . . . . . . . . . . 8.2 Macroscopic constraint equations for two species, strong interactions 8.3 Energy inequalities for two species, strong interaction . . . . . . . . 9 Approximate macroscopic equations . . . . . . . . . . . . . . 9.1 Approximate conservation of mass, momentum and energy for one species . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Conservation defects . . . . . . . . . . . . . . . . . 9.1.2 Decomposition of flux terms . . . . . . . . . . . . 9.1.3 Decomposition of acceleration terms . . . . . . . . 9.2 Approximate conservation of mass, momentum and energy for two species . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Conservation defects . . . . . . . . . . . . . . . . . 9.2.2 Decomposition of flux terms . . . . . . . . . . . . 9.2.3 Decomposition of acceleration terms . . . . . . . . 9.2.4 Proof of Propositions 9.5 and 9.6 . . . . . . . . . . 9.2.5 Proofs of Lemmas 9.7, 9.8, 9.9 and 9.10 . . . . . .

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278 287 290 293 295 299

10 Acoustic and electromagnetic waves . . . . . . . . . . . . . . . . . . . . 313 10.1 Formal filtering of oscillations . . . . . . . . . . . . . . . . . . . . . 314 10.2 Rigorous filtering of oscillations . . . . . . . . . . . . . . . . . . . . 318 11 Grad’s moment method . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Weak convergence of fluctuations, collision integrands and electromagnetic fields . . . . . . . . . . . . . . . . . . 11.1.2 Constraint equations, Maxwell’s system and the energy inequality . . . . . . . . . . . . . . . . . . 11.1.3 Evolution equations . . . . . . . . . . . . . . . . . . . . . 11.1.4 Temporal continuity, initial data and conclusion of the proof 12 The renormalized relative entropy method . . . . . . . . . . . . 12.1 The relative entropy method: old and new . . . . . . . . . . . 12.2 Proof of Theorem 4.6 on weak interactions . . . . . . . . . . . 12.2.1 Weak convergence of fluctuations, collision integrands and electromagnetic fields . . . . . . . . . . . . . . . . 12.2.2 Constraint equations, Maxwell’s system and the energy inequality . . . . . . . . . . . . . . . . 12.2.3 The renormalized modulated entropy inequality . . . . 12.2.4 Convergence and conclusion of the proof . . . . . . . . 12.3 Proof of Theorem 4.7 on strong interactions . . . . . . . . . . 12.3.1 Weak convergence of fluctuations, collision integrands and electromagnetic fields . . . . . . . . . . . . . . . .

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12.3.2 Constraint equations, Maxwell’s system and the energy inequality . . . . . . . . . . . . . . . . . . . . 359 12.3.3 The renormalized modulated entropy inequality . . . . . . . . 360 12.3.4 Convergence and conclusion of the proof . . . . . . . . . . . . 380 Appendix A: Cross-section for momentum and energy transfer . . . . . . . 385 Appendix B: Young inequalitites . . . . . . . . . . . . . . . . . . . . . . . . 389 Appendix C: End of proof of Lemma 7.7 on hypoelliptic transfer of compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Part I

Formal derivations and macroscopic weak stability

Chapter 1

The Vlasov–Maxwell–Boltzmann system In the present monograph, we intend to investigate in a rather systematic way the scaling limits of the Vlasov–Maxwell–Boltzmann system 8 q ˆ @t f C v rx f C .E C v ^ B/ rv f D Q.f; f /; ˆ ˆ m ˆ ˆ ˆ ˆ (Vlasov–Boltzmann) ˆ ˆ ˆ Z ˆ ˆ ˆ ˆ 0 0 @t E rot B D 0 q f v dv; ˆ ˆ ˆ R3 ˆ ˆ ˆ ˆ (Ampère) ˆ ˆ < @t B C rot E D 0; (1.1) ˆ ˆ (Faraday) ˆ ˆ Z ˆ ˆ ˆ q ˆ ˆ div E D f dv 1 ; ˆ ˆ 0 ˆ R3 ˆ ˆ ˆ ˆ (Gauss) ˆ ˆ ˆ ˆ ˆ div B D 0; ˆ ˆ : (Gauss)

leading to viscous incompressible magnetohydrodynamics, and to justify rigorously the corresponding asymptotics. More precisely, the Vlasov–Maxwell–Boltzmann system describes the evolution of a gas of one species of charged particles (cations and anions (or electrons), i.e., positively and negatively charged ions, respectively) of mass m > 0 and charge q 2 R, subject to self-induced electromagnetic forces. Such a gas of charged particles, under a global neutrality condition, is called a plasma. The particle number density f .t; x; v/ 0, where t 2 Œ0; 1/, x 2 R3 and v 2 R3 , represents the distribution of particles which, at time t, are at position x and have velocity v. The evolution of the density f is governed by the Vlasov–Boltzmann equation, which is the first line of (1.1). In essence, it tells that the variation of the density f along the trajectories of the particles (represented by the transport term @t f C v rx f ) is subject to the influence of a Lorentz force q .E C v ^ B/ (represented q by the Vlasov term m .E C v ^ B/ rv f ) and inter-particle collisions in the gas (represented by the Boltzmann collision operator Q.f; f /). The Lorentz force acting on the gas is self-induced. That is, the electric field E.t; x/ and the magnetic field B.t; x/ are generated by the motion of the particles in the plasma itself. Their evolution is governed by Maxwell’s equations, which are the remaining lines of (1.1), namely Ampère’s equation, Faraday’s equation and Gauss’ laws. Here, the physical constants 0 ; 0 > 0 are, respectively, the vacuum per-

4

1 The Vlasov–Maxwell–Boltzmann system

meability (or magnetic constant) and the vacuum permittivity (or electric constant). Recall that the speed of light is determined by the formula c D p10 0 . We will also consider the two-species Vlasov–Maxwell–Boltzmann system 8 qC ˆ ˆ ˆ @t f C C v rx f C C C .E C v ^ B/ rv f C D Q.f C ; f C / C Q.f C ; f /; ˆ ˆ m ˆ ˆ ˆ ˆ (Vlasov–Boltzmann for cations) ˆ ˆ ˆ ˆ q ˆ ˆ @t f C v rx f .E C v ^ B/ rv f D Q.f ; f / C Q.f ; f C /; ˆ ˆ ˆ m ˆ ˆ ˆ ˆ (Vlasov–Boltzmann for anions) ˆ ˆ Z ˆ ˆ C C ˆ ˆ ˆ 0 0 @t E rot B D 0 q f q f v dv; ˆ < R3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

(Ampère)

@t B C rot E D 0; div E D

1 0

Z

(Faraday)

C C q f q f dv;

R3

(Gauss)

div B D 0; (Gauss)

(1.2) which is more physically accurate, since it describes the evolution of a gas of two species of oppositely charged particles (cations of charge q C > 0 and mass mC > 0, and anions of charge q < 0 and m > 0), subject to self-induced electromagnetic forces. Thus, the particle number density f C .t; x; v/ 0 represents the distribution of the positively charged ions (i.e., cations), while the particle number density f .t; x; v/ 0 represents the distribution of the negatively charged ions (i.e., anions). Note that the collision operators Q.f C ; f / and Q.f ; f C / have been added to the right-hand sides of the respective Vlasov–Boltzmann equations in (1.2) in order to account for the variations in the densities f C and f due to interspecies collisions. We refer to [64] for a discussion on the validity of such systems from a physical viewpoint. Henceforth, for the mere sake of mathematical convenience, we will make the simplification that both kinds of particles have the exact same mass m˙ D m > 0 and charge q ˙ D q > 0. Even though this reduction may first appear rather unphysical, it remains nevertheless a reasonable approximation since the masses of cations and anions only differ by the mass of a few electrons, which is several orders of magnitude less than that of atomic nuclei. Anyway, we believe that the essential mathematical difficulties are contained in this case, and we expect that most of the analysis contained in this work carries over to the case of distinct masses, as long as they remain

1.1 The Boltzmann collision operator

5

of a comparable order of magnitude. We refer to [46] for an independent formal study of some hydrodynamic limits of the two-species Vlasov–Maxwell–Boltzmann system including the case of unequal masses, leading in particular to a formal justification of the Hall effect, which we will not address here. The mathematical framework we shall consider is the one defined by physical a priori estimates, namely entropy and energy bounds, which corresponds to renormalized or even weaker solutions of the Vlasov–Maxwell–Boltzmann systems. For the sake of simplicity, we will consider, throughout this work, that the spatial domain is in fact the whole space D R3 , thus avoiding the complicated discussion of boundary conditions. The general strategy that will be used to study magnetohydrodynamic limits is therefore based on uniform a priori bounds, weak compactness, and either the moment method of Grad, or some generalized relative entropy method, which are the only way to deal with very weak solutions: strong convergence requires indeed – at least – that local conservation laws are satisfied.

1.1 The Boltzmann collision operator The Boltzmann collision operator, present in the right-hand side of the Vlasov– Boltzmann equations in (1.1) and (1.2), is the quadratic form, acting on the velocity variable, associated to the bilinear operator Z Z 0 0 Q.f; h/ D (1.3) f h f h b.v v ; /d dv ; R3

S2

where we have used the standard abbreviations f D f .v/;

f 0 D f .v 0 /;

h D h.v /;

h0 D h.v0 /;

with .v 0 ; v0 / given by v0 D

v C v jv v j C ; 2 2

v0 D

v C v jv v j ; 2 2

and b.v v ; / 0 is the collision kernel, whose properties are detailed below. One can easily show that the quadruple .v; v ; v 0 ; v0 / parametrized by 2 S2 provides the family of all solutions to the system of four equations v C v D v 0 C v0 ; jvj2 C jv j2 D jv 0 j2 C jv0 j2 :

(1.4)

At the kinetic level, these relations express the fact that interparticle collisions are assumed to be elastic and thus conserve momentum and energy, where .v; v / denote

6


the pre-collisional velocities and .v 0 ; v0 / denote the post-collisionalvelocities of two vv interacting particles. Notice that the transformation .v; v ; / 7! v 0 ; v0 ; jvv is j involutional. It is to be emphasized that the definition of the Boltzmann operator for interspecies collisions with distinct masses is more complex. Indeed, in this case, the microscopic conservations of momentum and energy are 0 0 C m v ; mC vC C m v D mC vC 0 2 0 2 j C m jv j : mC jvC j2 C m jv j2 D mC jvC

Therefore, the masses must appear in the convolution relations defining the mixed collision operators, which become highly singular whenever the mass ratio tends to infinity or to zero. Again, for mathematical convenience, we will not deal with this case and stick to equal masses. The Boltzmann collision operator can therefore be split, at least formally, into a gain term and a loss term: Q.f; h/ D QC .f; h/ Q .f; h/ Z Z 0 0 f h b dv d D R3 S2

R3 S2

f h b dv d:

The loss term counts all collisions in which a given particle of velocity v will encounter another particle, of velocity v , and thus will change its velocity leading to a loss in the number of particles of velocity v, whereas the gain term measures the number of particles of velocity v which are generated due to some collision between particles of velocities v 0 and v0 , i.e., it counts the events where two particles with velocities v 0 and v0 collide and give rise to two particles with velocities v and v after the collision. The collision kernel b D b.z; /, where .z; / 2 R3 S2 , present in the integrand of (1.3), is a measurable function positive (b > 0) almost everywhere, which somehow measures the statistical repartition of post-collisional velocities .v 0 ; v0 / given the pre-collisional velocities .v; v /. It is directly related to an important physical quantity called the cross-section (or the differential cross-section; see [20, 76]) so that, by common abuse of language, we will often refer to the collision kernel b.z; / as the cross-section. Its precise form depends crucially on the nature of the microscopic interactions, thus, it is determined by the intermolecular forces that are being considered. However, due to the Galilean invariance of collisions, it only depends on the magnitude of the relative velocity jzj and on the deviation angle , or deflection z (scattering) angle, defined by cos D k where k D jzj . We will therefore sometimes abuse notation and write b.z; / D b.jzj; cos / without any confusion since the arguments of b are then either vectors or scalars. It is a common mathematical simplification, called the cutoff assumption, to suppose that the cross-section is at least locally integrable, i.e., that b.z; / belongs to L1loc .R3 S2 /. However, this hypothesis fails to hold when long-range interaction

1.1 The Boltzmann collision operator

7

forces are present between the particles in the gas. Thus, in this non-cutoff case, the collision kernel is non-integrable. This is due to a strong singularity of the kernel in the angular variable created by the enormous amount of grazing collisions in the gas, i.e., collisions whose deflection angle is almost null. For instance, if the particles are assumed to interact via a given repulsive potential ˆ.r/, where r > 0 denotes the distance between two interacting particles, then the post-collisional velocities and especially the deviation angle can be computed in terms of the impact parameter ˇ, i.e., the distance of closest approach if the particles were not to interact, and the relative velocity z D v v as the result of a classical scattering problem (see [20] for instance): Z .ˇ; z/ D 2 0

ˇ s0

12 4 ˇ 1 u 2ˆ du; jzj u 2

where s0 is the positive root of 1

ˆ.s0 / ˇ2 4 D 0: 2 jzj2 s0

Then the cross-section b is implicitly defined by b.jzj; cos / D

ˇ @ˇ jzj: sin @

It can be made fully explicit in the case of hard spheres b.jzj; cos / D a2 jzj; where a > 0 is the (scaled) radius of the spheres. As shown by Maxwell, it is possible to obtain a rather explicit expression for a wide class of physically relevant collision kernels (see [76] and references therein), namely the so-called inverse power kernels. This terminology stems from the fact that these kernels model a gas whose particles interact according to an inverse power 1 potential ˆ.r/ D r s1 , where r > 0 represents the distance between two particles and s > 2. Maxwell’s calculations show that in such a case one has b .jzj; cos / D jzj b0 .cos /;

D

s5 ; s1

where the angular cross-section b0 .cos / is smooth on 2 .0; / and has a nonintegrable singularity at D 0 behaving as b0 .cos / sin

1 1C

;

D

2 ; s1

where the factor sin accounts for the Jacobian determinant of spherical coordinates. Notice that, in this particular situation, b.z; / is thus not locally integrable, which

8


is not due to the specific form of inverse power potential. In fact, one can show (see [76]) that a non-integrable singularity arises if and only if forces of infinite range are present in the gas. The case of Maxwellian molecules s D 5 corresponds to D 0, which is not physically relevant, but enables one to perform many explicit calculations in agreement with physical observations. It is customary to loosely classify crosssections into two categories: hard and soft, respectively corresponding to the superMaxwellian (s > 5) and the sub-Maxwellian cases (s < 5). We will however not employ this dichotomy, since our hypotheses will allow us to treat all hard and soft kernels in a single unified theory. It turns out that the limiting case s D 2, which corresponds to Coulombian interactions, is not well suited for Boltzmann’s equation, as the Boltzmann collision operator should be replaced by the Landau operator in order to handle that situation (see [76]). The other limiting case s D 1 corresponds formally to the hard spheres case.

1.2 Formal macroscopic properties Using the well-known facts (see [21]) that transforming .v; v ; / 7! .v ; v; / and vv .v; v ; / 7! v 0 ; v0 ; jvv merely induces mappings with unit Jacobian determij nants, known as the pre-post-collisional changes of variables or simply collisional symmetries, one can show that Z Q.f; f /.v/'.v/ dv R3 Z (1.5) 0 0 1 0 0 f f ff b.v v ; / ' C ' ' ' dvdv d; D 4 R3 R3 S2 for all f .v/ and '.v/ regular enough. It then follows that the above integral vanishes, for every suitable f .v/, if and only if '.v/ is a collision invariant, i.e., a solution of the equation ' C ' ' 0 '0 D 0: ˚ Clearly, according to (1.4), any linear combination of 1; v1 ; v2 ; v3 ; jvj2 is a collision invariant. Moreover, it can be shown, under rather weak assumptions, that these are the only possible collision invariants (see [21, Section 3.1]). Thus, successively multiplying the Vlasov–Boltzmann equation in (1.1) by the collision invariants and then integrating in velocity yields formally the local conservation laws 0 1 0 0 1 1 Z Z Z 1 v 0 q @t f @ v A dv C rx f @v ˝ v A dv D f @E C v ^ B A dv; 2 2 3 3 m jvj jvj R R R3 Ev v 2 2 (1.6)

9

1.2 Formal macroscopic properties

which provide the link to a macroscopic description of the gas. In the case of two species (1.2), we obtain (recall that we are assuming equal masses m˙ D m and charges q ˙ D q) Z Z f ˙ dv C rx f ˙ v dv D 0 (1.7) @t R3

R3

and Z C v˝v 2 dv dv C rx f Cf jvj R3 R3 2 2 v Z C E Cv^B q D f f dv: Ev m R3

Z @t

C f Cf

v

jvj2

(1.8)

On the other hand, the standard energy estimates for Maxwell’s system in (1.1) and (1.2) (we refer to [45] for more details on Maxwell’s equations) are obtained, first, by taking the scalar product of the Ampère and Faraday equations with E and B, respectively, and summing the resulting quantities, which yields the conservation laws for one species Z 0 0 jEj2 C jBj2 @t f E v dv; (1.9) C rx .E ^ B/ D 0 q 2 R3 and for two species Z C 0 0 jEj2 C jBj2 C rx .E ^ B/ D 0 q f f E v dv: (1.10) @t 2 R3 Second, by taking the vector product of the Ampère and Faraday equations with B and E, respectively, employing Gauss’ laws when necessary, and summing the resulting quantities, which yields the conservation laws for one species 0 0 jEj2 C jBj2 0 0 @t .E ^ B/ C rx rx .0 0 E ˝ E C B ˝ B/ 2 Z D 0 q f .E C v ^ B/ dv C 0 qE; R3

(1.11)

and for two species 0 0 jEj2 C jBj2 0 0 @t .E ^ B/ C rx rx .0 0 E ˝ E C B ˝ B/ 2 Z C D 0 q f f .E C v ^ B/ dv: R3

(1.12)

10


Notice the similitude of the source terms in (1.6), (1.9), (1.11), and in (1.8), (1.10), (1.12). The other very important feature of the Boltzmann equation comes also from the symmetries of the collision operator. Without caring about integrability issues, we plug ' D log f into the symmetrized integral (1.5) and use the properties of the logarithm to find Z def D.f / D Q.f; f / log f dv 3 0 0 ZR 0 0 1 f f b.v v ; / dvdv d 0: D f f ff log 4 R3 R3 S2 ff (1.13) R The so defined entropy dissipation R3 D.f /.t; x/dx is non-negative and the funcRt R tional 0 R3 D.f /.s; x/dxds is therefore non-decreasing on t > 0. This leads to Boltzmann’s H -theorem, also known as the second principle of thermodynamics, stating that the entropy Z f log f dv R3

is (at least formally) a Lyapunov functional for the Boltzmann equation. Indeed, formally multiplying the Vlasov–Boltzmann equation in (1.1) by log f and then integrating in space and velocity clearly leads to Z Z d f log f .t; x; v/ dvCrx f log f .t; x; v/v dvCD.f /.t; x/ D 0: (1.14) dt R3 R3 A similar procedure on the two-species Vlasov–Boltzmann equations in (1.2) yields Z C d f log f C C f log f .t; x; v/ dv dt R3 Z C (1.15) C rx f log f C C f log f .t; x; v/v dv R3 C D f C C D .f / C D f C ; f .t; x/ D 0; where we have denoted the mixed entropy dissipation Z def .Q.f; h/ log f C Q.h; f / log h/ dv D .f; h/ D R3 0 0 Z 0 0 1 f h b.v v ; / dvdv d 0: D f h f h log 2 R3 R3 S2 f h (1.16) As for the equation Q.f; f / D 0, it is possible to show, since necessarily D.f / D 0 in this case, that it is only satisfied by the so-called Maxwellian distributions MR;U;T

1.2 Formal macroscopic properties

11

defined by MR;U;T .v/ D

R

e 3

jvU j2 2T

;

.2T / 2

where R 2 RC , U 2 R3 and T 2 RC are respectively the macroscopic density, bulk velocity and temperature, under some appropriate choice of units. The relation Q.f; f / D 0 expresses the fact that collisions are no longer responsible for any variation in the density and so, that the gas has reached statistical equilibrium. In fact, it is possible to show that if the density f is a Maxwellian distribution for some R.t; x/, U.t; x/ and T .t; x/, then the macroscopic conservation laws (1.6) turn out to constitute a compressible Euler system with electromagnetic forcing terms. Similarly, for two species of particles,if the plasma thermodynamic C reaches equi C C librium so that the relations Q f ; f ; f C Q f D 0 and Q C f ;f C C CD f C Qf ; f D 0 are satisfied simultaneously, then necessarily D f D f C ; f D 0, which implies that f C D MRC ;U C ;T C and f D MR ;U ;T with U C D U and T C D T , but not necessarily equal masses. In this case, it is possible to show that the macroscopic system of conservation laws (1.7)–(1.8) constitute a compressible Euler system with electromagnetic forcing terms. Finally, we define the (global) relative entropy, for any particle number density f 0 and any Maxwellian distribution MR;U;T , by Z f f log f C MR;U;T .t/ dxdv 0: H f jMR;U;T .t/ D MR;U;T R3 R3 (1.17) We will more simply denote the relative entropy by H.f /, whenever the relative Maxwellian distribution is clearly implied. The global control of the relative entropies then follows from the non-negativity of the entropy dissipation. Indeed, combining the H -theorem (1.14) with the global conservation of mass and energy from (1.6) and Maxwell’s energy conservation (1.9), it is in general possible to establish for one species (see [32], for instance), further integrating in time and space, by virtue of the convexity properties of the entropies and the entropy dissipations, the following weaker relative entropy inequality, for any t > 0, Z f f log f C M .t/ dxdv M R3 R3 Z Z tZ 0 1 1 2 2 jEj C jBj .t/ dx C D.f /.s/ dxds C 2 R3 m m0 R3 0 (1.18) Z f in in in f log f C M dxdv M R3 R3 Z 1 1 0 in 2 C jB in j2 dx; jE j C 2 R3 m m0 where f in ; E in ; B in denotes the initial data and M denotes a global normalized

12


Maxwellian distribution, M D M1;0;1 D

1 .2/

3 2

e

jvj2 2

:

Similarly, for two species, combining the H -theorem (1.15) with the global conservation of mass and energy from (1.7)–(1.8) and Maxwell’s energy conservation (1.10), we get the entropy inequality, for all t > 0, Z fC f f C log f C C M .t/ C f log f C M .t/ dxdv M M R3 R3 Z 1 1 0 C jBj2 .t/ dx jEj2 C 3 2 R m m0 Z tZ C

C D f C D .f / C D f C ; f .s/ dxds R3 0 Z f Cin f Cin log f Cin C M dxdv M R3 R3 Z f in in in f log C M dxdv f C M R3 R3 Z 1 1 0 in 2 C jB in j2 dx; jE j C 2 R3 m m0 (1.19) where f Cin ; f in ; E in ; B in denotes the initial data. Generally speaking, the H -theorem and the entropy inequalities (1.18) and (1.19) together with the conservation laws (1.6) and (1.7)–(1.8) constitute key elements in the study of hydrodynamic limits.

1.3 The mathematical framework The construction of suitable global solutions to the Vlasov–Maxwell–Boltzmann system (1.1) 8 @t f C v rx f C .E C v ^ B/ rv f D Q.f; f /; ˆ ˆ Z ˆ ˆ ˆ ˆ ˆ E rot B D f v dv; @ t ˆ ˆ < R3 @t B C rot E D 0; Z ˆ ˆ ˆ ˆ ˆ div E D f dv 1; ˆ ˆ ˆ R3 ˆ : div B D 0;

1.3 The mathematical framework

13

or to the two-species Vlasov–Maxwell–Boltzmann system (1.2) 8 @t f ˙ C v rx f ˙ ˙ .E C v ^ B/ rv f ˙ D Q.f ˙ ; f ˙ / C Q.f ˙ ; f /; ˆ ˆ Z ˆ ˆ C ˆ ˆ ˆ E rot B D f f v dv; @ t ˆ ˆ < R3 @t B C rot E D 0; ˆ Z ˆ C ˆ ˆ ˆ div E D f f dv; ˆ ˆ ˆ R3 ˆ : div B D 0; for large initial data is considered of outstanding difficulty, due to the lack of dissipative phenomena in Maxwell’s equations, which are hyperbolic. Here, for the sake of simplicity, we have discarded all free parameters, since these are irrelevant for the existence theory. Thus, so far, the only known answer to this problem is due to Lions in [55], where a rather weak notion of solutions was derived: the so-called measurevalued renormalized solutions. However, these solutions failed to reach mathematical consensus on their usefulness due to their very weak aspect. It should be mentioned that an alternative approach yielding strong solutions, provided smallness and regularity assumptions on the initial data are satisfied, was obtained more recently by Guo in [41]. But such solutions fall out of the scope of our derivation of hydrodynamic limits since they are not based on the physical entropy and energy estimates. Anyway, were we to consider such strong solutions, our approach and strategy would remain strictly the same, for, as we are about to see in Chapter 2 below, the only uniform bounds valid in the hydrodynamic limit are precisely the physical entropy and energy estimates. This poor understanding of the mathematical theory of the Vlasov–Maxwell– Boltzmann system is the reason why getting rigorous convergence results is so complex. For the sake of readability, we have therefore decided to separate the different kinds of difficulties. In Part II, we will prove conditional convergence results restricting our attention to the case of Maxwellian cross-sections, i.e., b 1, for mere technical simplicity, and assuming the existence of renormalized solutions to (1.1) and (1.2), which is actually not known. It is to be emphasized that, even if this notion of solution is relatively rough, the convergence proof in this weak case has no purely technical difficulty specific to this roughness. Indeed, were we to deal with stronger solutions, the strategy of proof would not be any different or easier, because we are considering here only the uniform bounds which come from physical estimates. In this framework, we can focus on the key arguments of the convergence proof, which are not so different from the ones used for hydrodynamic limits of neutral gases. A crucial point is to understand how to get strong compactness on macroscopic fields, which cannot be dealt with using L1 mixing lemmas such as in [37] because of the electromagnetic forcing terms. We will thus first prove strong compactness with

14


respect to velocity, and then use refined hypoelliptic estimates established in [7] in order to transfer the strong compactness to the spatial variable (Chapter 7). The other key point which requires a specific treatment is the study of fast timeoscillations insofar as they possibly couple weak compressibility with strong electromagnetic effects (Chapter 10). The Parts III and IV will be then devoted to the understanding of additional technical difficulties related to the fact that we are not able to build renormalized solutions to the Vlasov–Maxwell–Boltzmann systems, but only even weaker solutions. In the case of singular collision kernels, using the regularizing properties of the collision operator with respect to v, we will actually show the existence of renormalized solutions with a defect measure in the sense of Alexandre and Villani. The major change is the fact that the renormalized kinetic equation is replaced by an inequality (the consistency coming from the conservation of mass). This leads to the introduction of a defect measure. The important new step of the convergence proof is then to establish that this defect measure vanishes in the fast relaxation limit, which comes from refined entropy dissipation estimates. There are also many additional technical steps due to the singularity of the collision kernel, which makes the control of the conservation defects and the hypoelliptic transfer of compactness more difficult. In the apparently simpler case of cutoff collision kernels, because of the lack of strong compactness estimates, we are not able to prove that approximate solutions fN to the Vlasov–Maxwell–Boltzmann systems (1.1) and (1.2) converge to actual renormalized solutions. Indeed, without strong compactness properties, it is not possible to establish that ˇ.fN / ! ˇ.f / for any renormalization ˇ, which accounts for the introduction of Young measures and the definition of a very rough notion of solution, namely the measure-valued renormalized solutions. Of course, the physical meaning of such weak solutions is unclear, which probably explains why they have not been studied so far. Nevertheless, we will establish here that – in the fast relaxation limit – they exhibit the expected behavior, converging to the relevant magnetohydrodynamic model, which can be considered as an indication of their physical relevance. The key point of the proof will be to obtain integrated versions of all estimates with respect to the Young measures, and to prove that asymptotically the Young measures are not seen by the limiting equation, even though they do not converge to Dirac masses due to lack of uniqueness of solutions in the limiting systems.

Chapter 2

Scalings and formal limits In view of what is known on hydrodynamic limits of the Boltzmann equation (see [70] and the references therein), which corresponds to the particular case where particles are not charged, i.e., q D 0 in (1.1), we will focus on incompressible diffusive regimes, since we do not expect to be able to obtain a complete mathematical derivation for other choices of scalings.

2.1 Incompressible viscous regimes In the absence of an electromagnetic field, the Boltzmann equation can be rewritten in non-dimensional variables (see [70, Section 2.2]) St@t f C v rx f D

1 Q.f; f /; Kn

where we have introduced the following parameters: the Knudsen number Kn D l00 , measuring the ratio of the mean free path 0 to the observation length scale l0 ; the Strouhal number St D c0l0t0 , measuring the ratio of the observation length scale l0 to the typical length c0 t0 run by a particle during a unit of time t0 , where c0 is the speed of sound (or thermal speed); choosing the length l0 , time t0 and velocity scales u0 in such a way that we observe a macroscopic motion, i.e., u0 D lt00 , we have the identity St D Ma where the Mach number Ma D uc00 is defined as the ratio of the bulk velocity u0 to the thermal speed. Hydrodynamic approximations are obtained in the fast relaxation limit Kn ! 0, which precisely corresponds to the asymptotic regime where the fluid under consideration satisfies the continuum hypothesis, for the mean free path becomes infinitesimally small. Because of the von Kármán relation for perfect gases, we then expect the flow to be dissipative when the Reynolds number Re

Ma ; Kn

measuring the inverse kinematic viscosity of the gas, is of order 1, i.e., when the Mach number also tends to 0.

16

2 Scalings and formal limits

In order to ensure the consistency of these scaling assumptions, we will consider – as usual – data which are fluctuations g of order Ma f D M.1 C Ma g/; around a global normalized Maxwellian equilibrium M.v/ D

1 .2/

3 2

e

jvj2 2

;

of density 1, bulk velocity 0 and temperature 1. Thus, as is well-known since the works of Bardos, Golse and Levermore [10, 11], the viscous incompressible hydrodynamic regimes of collisional kinetic systems are obtained in the fast relaxation limit when the above-mentioned dimensionless numbers Kn, St and Ma, are all of the same order > 0, say. In the sequel, we will therefore restrict our attention to the scaled Vlasov–Maxwell–Boltzmann system 8 ql0 ˆ ˆ .E C c0 v ^ B/ rv f @t f C v rx f C ˆ 2 ˆ mc ˆ 0 ˆ ˆ ˆ ˆ f ˆ ˆ ˆ ˆ ˆ < c0 0 0 @t E rot B ˆ ˆ ˆ c0 @t B C rot E ˆ ˆ ˆ ˆ ˆ ˆ ˆ div E ˆ ˆ ˆ ˆ : div B

1 D Q.f; f /; D M .1 C g/ ; Z D 0 qc0 l0 f v dv; R3

(2.1)

D 0; Z ql0 f dv 1 ; D 0 R3 D 0;

and to the scaled two-species Vlasov–Maxwell–Boltzmann system 8 ql0 ˆ ˆ .E C c0 v ^ B/ rv f ˙ @t f ˙ C v rx f ˙ ˙ ˆ 2 ˆ mc ˆ 0 ˆ ˆ ˆ ˆ ˆ 1 ı2 ˆ ˙ ˙ ˆ D ; f / C Q.f Q.f ˙ ; f /; ˆ ˆ ˆ ˆ ˆ ˆ ˆ f ˙ D M 1 C g ˙ ; < Z C ˆ @ E rot B D qc l f f v dv; c 0 0 0 t 0 0 0 ˆ ˆ ˆ R3 ˆ ˆ ˆ c0 @t B C rot E D 0; ˆ ˆ ˆ Z ˆ ˆ C ql0 ˆ ˆ ˆ f f dv; div E D ˆ ˆ 0 R3 ˆ ˆ : div B D 0; (2.2)

2.2 Scalings for the electromagnetic field

17

where we have introduced another bounded parameter ı > 0 in front of the interspecies collision operator to differentiate the strength of interactions. The size of the parameter ı will be compared to the Knudsen number Kn D and we will distinguish three cases, due to their distinct asymptotic behavior: ı 1, strong interspecies interactions; ı D o.1/ and

ı

unbounded, weak interspecies interactions;

ı D O./, very weak interspecies interactions. Notice also that we have performed the same non-dimensionalization on the whole Vlasov–Maxwell–Boltzmann systems, which explains the presence of the parameters , c0 and l0 in Maxwell’s equations.

2.2 Scalings for the electromagnetic field As previously mentioned, we know from [10, 11] that the above systems (2.1) and (2.2) are expected to yield, at least formally, viscous incompressible hydrodynamic regimes, in the limit ! 0, and that the entropy inequalities (1.18) and (1.19) guarantee that the thermodynamic variables remain bounded and of leading order. Depending on the asymptotic behavior of the remaining physical constants, this may not yet be the case for the electromagnetic field, though. Our aim is to obtain a classification of all possible regimes in terms of (a minimal number of) non-dimensional parameters. Therefore, we proceed now to a dimensional analysis of the electromagnetic variables and make sure, through an appropriate change of units, that the relevant electromagnetic quantities remain bounded and of leading order, as ! 0. First, from (1.18), we get the scaled entropy inequality for the one-species system (2.1), for all t > 0, Z 1 f f log f C M .t/ dxdv 2 R3 R3 M Z Z Z 1 t 1 1 0 C 2 2 jBj2 .t/ dx C 4 D.f /.s/ dxds jEj2 C m0 0 R3 2c0 R3 m Z 1 f in in in 2 f log f C M dxdv R3 R3 M Z 1 1 0 in 2 in 2 C 2 2 jB j dx; jE j C m0 2c0 R3 m (2.3) where f in ; E in ; B in denotes the initial data.

18


For the two-species system (2.2), from (1.19), we get the scaled entropy inequality, for all t > 0, Z fC f 1 C C f log f C M .t/ C f log f C M .t/ dxdv 2 R3 R3 M M Z 0 1 1 C 2 2 jBj2 .t/ dx jEj2 C m0 2c0 R3 m Z Z C 1 t C 4 D f C D .f / C ı2 D f C ; f .s/ dxds 0 R3 Z f Cin 1 f Cin log 2 f Cin C M dxdv R3 R3 M Z f in 1 in in f log C M dxdv f C 2 R3 R3 M Z 1 1 0 in 2 in 2 C 2 2 jB j dx; jE j C m0 2c0 R3 m (2.4) where f Cin ; f in ; E in ; B in denotes the initial data. Note that the entropy inequalities (2.3) and (2.4) are the only uniform controls we have on the particle number densities and on the electric and magnetic fields, meaning that whatever the repartition of the free energy at the initial time, all the contributions are expected to be of the same order. Thus, up to a change of units in E and B, namely setting r 1 0 1 B; E; BQ D EQ D p c0 m c0 m0 so that EQ and BQ are uniformly controlled by the scaled entropy inequalities (2.3) or (2.4), we have (dropping the tildes for the sake of readability), for one species, 8 1 ˆ ˆ @t f C v rx f C .˛E C ˇv ^ B/ rv f D Q.f; f /; ˆ ˆ ˆ ˆ ˆ ˆ f D M .1 C g/ ; ˆ ˆ ˆ Z ˆ ˆ ˇ ˆ < f v dv; @t E rot B D 2 R3 (2.5) ˆ ˆ @ B C rot E D 0; ˆ t ˆ ˆ Z ˆ ˆ ˛ ˆ ˆ f dv 1 ; div E D 2 ˆ ˆ ˆ R3 ˆ ˆ : div B D 0;

2.2 Scalings for the electromagnetic field

19

and, for two species, 8 1 ı2 ˆ ˙ ˙ ˙ ˙ ˙ ˆ Q.f Q.f ˙ ; f /; f C v r f ˙ .˛E C ˇv ^ B/ r f D ; f / C @ ˆ t x v ˆ ˆ ˆ ˆ ˆ ˆ f ˙ D M 1 C g ˙ ; ˆ ˆ Z ˆ ˆ C ˆ ˇ < f f v dv; @t E rot B D 2 R3 ˆ ˆ ˆ @t B C rot E D 0; ˆ ˆ Z ˆ ˆ C ˆ ˛ ˆ ˆ f div E D f dv; ˆ 2 ˆ R3 ˆ ˆ : div B D 0; (2.6) where there are only three free parameters left (other than and ı) to describe the qualitative behaviors of the systems, namely: ˛ D c

0

ql0 p , m0

measuring the electric repulsion according to Gauss’ law;

q

ˇ D ql0 m0 , measuring the magnetic induction according to Ampère’s law; p p D c0 0 0 D u0 0 0 , which is nothing else than the ratio of the bulk velocity to the speed of light. Notice that these parameters are naturally constrained by the relation ˇD

˛ :

We will impose some natural restrictions on the size of ˛, ˇ and . First of all, we p will require that D O.1/. Note, however, that an unbounded D c0 0 0 may seem physically unrealistic, since it corresponds to a regime where the thermal speed (i.e., the speed of sound) exceeds the speed of light. As usual, such situations should only be interpreted as asymptotic regimes where appropriate physical approximations are valid. Moreover, in the one-species case, we will demand that ˛ and ˇ are of order O./, so that electric and magnetic forces create bounded acceleration terms in the Vlasov–Boltzmann equation in (2.5). Situations where one of these parameters is large compared to are much more complicated. Indeed, we expect the Lorentz force to strongly penalize the system, leading asymptotically to some nonlinear macroscopic constraint that we are not able to deal with in the one-species case. Actually, as far as we know, there is no systematic mathematical method to investigate such problems of nonlinear singular perturbation. For instance, understanding the dynamo effect is a related question which remains challenging.

20


Thus, on the whole, for one species, we will consider bounded parameters ˛, ˇ and satisfying ˛ D O./;

ˇ D O./;

D O.1/

and

ˇD

˛ :

We will then distinguish two critical cases, namely (1) ˛ D , ˇ D , D , (2) ˛ D 2 , ˇ D , D 1, and will explain how all other cases can be easily deduced from the above, just eliminating lower order terms which are too small. The full range of parameters will be described later on by Figure 2.1 on page 33. For the moment, we merely emphasize that the above-mentioned critical cases correspond exactly to the vertices of the domain represented in Figure 2.1. For two species, the restrictions on the size of the parameters ˛, ˇ and are not so explicitly deduced by inspection of the system (2.6), except in the case ı D 2 O./, which lowers the order of the interspecies collision term ı Q.f ˙ ; f / in (2.6) and is thus analogous to the one-species case. However, when ı is unbounded, 2

the interspecies collision term ı Q.f ˙ ; f / becomes a singular perturbation and, as a matter of fact, the need of asymptotically bounded acceleration terms in the macroscopic laws associated with the Vlasov–Boltzmann equations in (2.6) leads us to require that ˛ D O./ and ˇ D O.ı/. Note that ˇ D O./ is not required in this case, which is in sharp contrast with the one-species case. Thus, on the whole, for two species, we will consider bounded parameters ˛, ˇ, and ı satisfying either ˛ D O./;

ˇ D O./;

D O.1/ and ˇ D

˛ ;

ˇ D O.ı/;

D O.1/ and ˇ D

˛ ;

when ı D O./, or ˛ D O./;

otherwise. We will then distinguish two critical cases, namely (1) ˛ D , ˇ D , D , (2) ˛ D 2 , ˇ D , D 1, when ı D O./, and (1) ˛ D , ˇ D ı, D ı, (2) ˛ D ı, ˇ D ı, D 1,

2.3 Formal analysis of the one-species asymptotics

21

when ı is unbounded (note that the latter two cases coincide when ı 1), and will explain how all other cases can be easily deduced from the above, just eliminating lower order terms which are too small. Thus, as for one species, when ı D O./, the full range of parameters will be described by Figure 2.1 on page 33. Furthermore, when ı 1, the range of parameters will be represented by Figure 2.2 on page 67, while the case ı D o.1/ with ı unbounded will be described by Figure 2.3 on page 69. Again, we merely emphasize, for the moment, that the above-mentioned critical cases correspond exactly to the vertices of the respective domains represented in Figures 2.1, 2.2 and 2.3.

2.3 Formal analysis of the one-species asymptotics We proceed now to the systematic derivation of all asymptotic systems associated with the viscous incompressible regimes of a plasma of one species of particles. As mentioned in the previous section, we indeed expect to obtain different electromagnetic forcings for the fluid depending on: the strength of the magnetic induction, the strength of the electric induction, the ratio of the bulk velocity to the speed of light. A complete summary of all limiting systems is given in Section 2.3.4 below. Thus, our starting point is the scaled system 8 1 ˆ ˆ @t f C v rx f C .˛E C ˇv ^ B / rv f D Q.f ; f /; ˆ ˆ ˆ ˆ ˆ ˆ D M .1 C g / ; f ˆ ˆ ˆ Z ˆ ˆ ˇ ˆ < g vM dv; @t E rot B D R3 ˆ ˆ @t B C rot E D 0; ˆ ˆ ˆ Z ˆ ˆ ˛ ˆ ˆ div E D g M dv; ˆ ˆ ˆ R3 ˆ ˆ : div B D 0; supplemented with some initial data satisfying Z in 2 1 in 1 H f C jE j C jBin j2 dx < 1; 2 2 R3

(2.7)

22


where H.fin / D H.fin jM /. In particular, the corresponding scaled entropy inequality, where t > 0, Z Z Z 1 1 1 t 2 2 jE j C jB j dx C 4 H f C D.f /.s/ dxds 2 2 R3 0 R3 Z in 2 1 in 1 jE j C jBin j2 dx; 2 H f C 2 R3 (2.8) guarantees that the solution will remain – for all non-negative times – a fluctuation of order around the global equilibrium M : f D M.1 C g /: Note that the kinetic equation in (2.7) can then be rewritten, in terms of the fluctuation g , as ˛ 1 @t g Cvrx g C.˛E Cˇv^B /rv g E v .1 C g / D Lg C Q.g ; g /; (2.9) where we denote

Lg D

1 .Q.Mg; M / C Q.M; Mg// M

and

Q.g; g/ D

1 Q.Mg; Mg/: M (2.10)

2.3.1 Thermodynamic equilibrium The entropy inequality (2.8) provides uniform bounds on E , B and g . Therefore, assuming some formal compactness, up to extraction of subsequences, one has E * E;

B * B;

g * g;

in a weak sense to be rigorously detailed in a subsequent chapter. Then, multiplying (2.9) by and taking formal limits as ! 0 shows that Lg D 0. It can be shown (see Proposition 5.5 below) that the kernel of the linL coincides earized Boltzmann operator ˚ exactly with the vector space spanned by the collision invariants 1; v1 ; v2 ; v3 ; jvj2 . Thus, we conclude that g is an infinitesimal Maxwellian, namely, a linear combination of collision invariants 2 3 jvj ; (2.11) g DCuvC 2 2 where 2 R, u 2 R3 and 2 R only depend on t and x, and are respectively the fluctuations of density, bulk velocity and temperature. The fact that the fluctuations assume the infinitesimal Maxwellian form describes that the gas reaches thermodynamic (or statistical) equilibrium, in the fast relaxation limit.


23

We define now the macroscopic fluctuations of density , bulk velocity u , and temperature by Z D

ZR

u D

3

g M dv;

g vM dv; 2 Z jvj g D 1 M dv; 3 R3 R3

and the hydrodynamic projection …g of g by …g D C u v C

jvj2 3 ; 2 2

which is nothing but the orthogonal projection of g onto the kernel of L in L2 .Mdv/. Note that the previous step establishing the convergence of g towards thermodynamic equilibrium yields, in fact, the uniform boundedness of 1 Lg , which implies, at least formally, that g …g D O./: (2.12) This convergence may also be derived directly from the uniform control of the entropy dissipation 14 D.f / in the entropy inequality (2.8), provided we can control the large values of the fluctuations. Indeed, according to (1.13), we write 0 0 Z 0 f0 f 1 f f f f f f log 1 C f f b dvdv d: D.f / D 4 R3 R3 S2 f f f f Therefore, since the non-negative function z log.1 C z/ behaves essentially as z 2 , for small values of jzj, we deduce a formal control on 1 4 4

Z R3 R3 S2

0 f f f0 f f f

2 f f b dvdv d:

Then, since 0 0 0 f f D g0 C g g g C 2 g0 g g g ; f0 f 0 g g is uniformly bounded, which, in other words, we infer that 1 g0 C g amounts to a control on 1 Lg . The asymptotic dynamics of . ; u ; / is then governed by fluid equations, to be obtained from the moment equations associated with (2.9). Thus, successively 2 multiplying (2.9) by the collision invariants 1, v and jvj2 , and integrating in Mdv,

24


yields 8 1 ˆ ˆ @t C div u D 0; ˆ ˆ ˆ Z ˆ < 1 1 ˛ ˇ ˛ g M dv; @t u C rx . C / 2 E D E C u ^ B div ˆ R3 ˆ Z ˆ ˆ ˆ 5 ˛ 1 3 ˆ : g M dv; @t . C / C div u D u E div 2 2 R3 (2.13) where the kinetic momentum and energy fluxes are defined by 2 jvj2 5 jvj .v/ D v ˝ v Id; .v/ D v: (2.14) 3 2 2 Recall that we are assuming ˛ D O./ and ˇ D O./. Hence, the nonlinear terms in the right-hand side of (2.13) containing the electromagnetic fields are expected to be bounded. Furthermore, notice that the polynomials .v/ and .v/ are 2 orthogonal to the collision R invariants in the L .Mdv/ inner product. That is to say, R ' M dv D 0 and ' M dv D 0, for all collision invariants '.v/. Since, R3 R3 according to (2.12), g converges towards an infinitesimal Maxwellian with a rate O./, it is therefore natural to expect, at least formally, that the terms Z Z 1 1 g M dv D .g …g / M dv; R3 R3 Z Z 1 1 g M dv D .g …g / M dv; R3 R3 in (2.13) are bounded and have a limit. More precisely, it can be shown that, in general, the linearized Boltzmann operator L is self-adjoint and Fredholm of index zero on L2 .Mdv/ (or a variant of it depending on the cross-section). Therefore, its range is exactly the orthogonal complement of its kernel. It follows that 2 L2 .Mdv/ and 2 L2 .Mdv/ belong to the range of L and, thus, that there are inverses Q 2 L2 .Mdv/ and Q 2 L2 .Mdv/ such that D L Q and D L Q; (2.15) which can be uniquely determined by the fact that they are orthogonal to the kernel of L (i.e., to the collision invariants). Consequently, the macroscopic system (2.13) can be recast as 8 1 ˆ ˆ @t C div u D 0; ˆ ˆ ˆ Z ˆ < 1 1 ˛ ˇ ˛ Q dv; Lg M @t u C rx . C / 2 E D E C u ^ B div ˆ R3 ˆ Z ˆ ˆ ˆ 1 ˛ 1 3 ˆ : Lg Q M dv; @t C div u D u E div 2 R3 (2.16)

25


where the terms 1 Lg will be expressed employing the Vlasov–Boltzmann equation (2.9). The above macroscopic system (2.16) is coupled with Maxwell’s equations for E and B : 8 ˇ ˆ ˆ @t E rot B D u ; ˆ ˆ ˆ ˆ < @ B C rot E D 0; t (2.17) ˛ ˆ ˆ div E D ; ˆ ˆ ˆ ˆ : div B D 0: A careful formal analysis of the whole coupled macroscopic system (2.16)–(2.17) will yield the asymptotic dynamics of .; u; ; E; B/.

2.3.2 Macroscopic constraints At leading order, the system (2.16)–(2.17) describes the propagation of acoustic ! r 3 ; u ; 2 and electromagnetic .E ; B / waves: 0 1 0 1 Bqu C Bqu C B C B C 3 C B 3 C @t B B 2 C C W B 2 C D O.1/; @ E A @ E A B B

(2.18)

where the wave operator W , containing the singular terms from (2.16)–(2.17) and defined explicitly below, is antisymmetric (with respect to the L2 .dx/ inner product) and therefore can only have purely imaginary eigenvalues. The semi-group generated by this operator may thus produce fast time-oscillations, which we are about to discuss briefly. (1) When 1 (so that ˛ D O. 2 /), we have 1 0 1 0 div 0 0 0 q C B1 1 2 B rx 0 rx 0 0C 3 C B q C: (2.19) W D B 1 2 B 0 div 0 0 0C 3 C B @ 0 0 0 0 0A 0 0 0 0 0 Thus, the singular perturbation creates only high-frequency acoustic waves. Consequently, averaging over fast time-oscillations as ! 0, we get the macroscopic constraints div u D 0;

C D 0;

(2.20)

26


respectively referred to as incompressibility and Boussinesq relations. These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.17) div E D 0; div B D 0: (2) When D o.1/ and ˛ D O. 2 /, we have 0 1 0 div q0 B1r 1 2 0 r B x 3 x q B B 1 2 W D B 0 div 0 3 B @ 0 0 0 0 0 0

0

0

0

0

1

C C C C 0 0 C: C 0 1 rotA 1 rot 0

(2.21)

Thus, the singular perturbation creates both high-frequency acoustic and electromagnetic waves. However, these waves remain decoupled and have a comparable frequency of oscillation if and only if . By averaging these fast time-oscillations as ! 0, we get the macroscopic constraints div u D 0; C D 0; rot B D 0; rot E D 0:

(2.22)

These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.17) div E D 0; div B D 0: Hence,

E D 0;

(3) When D o.1/ and 0

˛ 2

is unbounded, we have 1

0

B1r B x B W D B B 0 B @ 0 0

B D 0:

1

div

0 q

2 3

div ˛ Id 2 0

1

q0

2 r 3 x

0 0 0

0 ˛2 Id 0 0 1 rot

0

1

C C C C 0 C: C 1 rotA 0

(2.23)

0

Thus, the singular perturbation creates both high-frequency acoustic and electromagnetic waves, which are coupled. These waves may or may not have comparable frequency of oscillation. By averaging these fast time-oscillations as ! 0, we get the macroscopic constraints h˛ i div u D 0; rx . C / D E; (2.24) ˇ u; rot E D 0; rot B D


27

where we have denoted by ˛ and ˇ the respective limits of ˛ and ˇ as ! 0. As usual, when ˛ D o./, the weak Boussinesq relation rx .C / D 0 can be improved to the strong Boussinesq relation C D 0, assuming and enjoy enough integrability. These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.17) h˛ i ; div B D 0: div E D The exact nature of time oscillations produced by the system (2.18), in the limit ! 0, will be rigorously discussed, with greater detail, later on in Chapter 10.

2.3.3 Evolution equations The previous step shows that, since W is singular, the asymptotic dynamics of ! r 3 ; u ; ; E ; B 2 becomes constrained to the kernel Ker W as ! 0. Moreover, since W is antisymmetric, its range is necessarily orthogonal to its kernel. Therefore, in order to get the asymptotic evolution equations for ! r 3 ; u; ; E; B ; 2 it is natural to project the system (2.18) onto Ker W , which will rid us of all the singular terms in (2.18) and allow us to pass to the limit. In other words, we will obtain the limiting dynamics of the system (2.18) by testing it against functions in Ker W . We will denote by P W L2 .dx/ ! L2 .dx/ the Leray projector onto solenoidal vector fields and by P ? D Id P the projector onto the orthogonal complement, that is, P D 1 rot rot and P ? D 1 r div. (1) When 1, the kernel of W , defined in (2.19), is obviously determined by all ! r 3 0 0 0 0 0 ; u ; ;E ;B 2 that satisfy

div u0 D 0

and

0 C 0 D 0:

It is then readily seen that its orthogonal complement Ker W? is determined by all ! r 3Q Q Q ; E ; B Q ; uQ ; 2

28


such that P uQ D 0

and

3Q Q D 0: 2

Hence, projecting the system (2.16)–(2.17) onto Ker W? yields Z 8 1 ˛ ˇ ˛ ˆ Q dv D P ˆ Lg M E C E C u ^ B ; @t P u C P div < 2 R3 Z ˆ 1 ˛ 3 ˆ : @t C div Lg Q M dv D u E : 2 R3 (2.25) (2) When D o.1/ and ˛ D O. 2 /, the kernel of W , defined in (2.21), is obviously determined by all ! r 3 0 ; u0 ; 0 ; E 0; B 0 2 that satisfy div u0 D 0; 0 C 0 D 0; rot E0 D 0;

rot B0 D 0:

It is then readily seen that its orthogonal complement Ker W? is determined by all ! r 3Q Q Q Q ; uQ ; ; E ; B 2 such that P uQ D 0; P ? EQ D 0;

3Q Q D 0; 2 P ? BQ D 0:

Hence, projecting the system (2.16) onto Ker W? also yields the system (2.25). Moreover, in view of Gauss’ laws, the projection of Maxwell’s equations (2.17) onto Ker W? yields no useful information. (3) When D o.1/ and ˛2 is unbounded, the wave operator W is defined by (2.23). Notice then that Gauss’ laws from (2.17) are invariant under the action of the wave operator W . Consequently, it is enough to consider the restriction of W to electromagnetic fields which verify Gauss’ laws. It follows that the kernel of W is obviously determined by all ! r 3 0 0 0 0 0 ; u ; ;E ;B 2


29

which satisfy ˛ ˇ 0 u ; rx 0 C 0 D E0 ; ˛ 0 0 0 div B D 0: div E D ; rot B0 D

It is then readily seen that its orthogonal complement Ker W? is determined by all ! r 3Q Q Q Q ; uQ ; ; E ; B 2 such that

ˇ 3Q rot uQ C BQ D 0; Q D 0; 2 ˛ Q div BQ D 0: div E D Q ; Considering the magnetic potential BQ D rot AQ , uniquely determined if div AQ D 0 (i.e., fixing the Coulomb gauge), the above set of constraints can be recast as ˇ 3Q P uQ C AQ D 0; Q D 0; 2 ˛ div AQ D 0: div EQ D Q ; Hence, projecting the system (2.16) onto Ker W? yields Z 8 1 ˇ ˇ ˛ ˆ Q ˆ < @t P u C A C P div 3 Lg M dv D P E C u ^ B ; R Z ˆ 1 ˛ 3 ˆ : @t Lg Q M dv D u E ; C div 2 R3 (2.26) where B D rot A and div A D 0, and where we have used that Faraday’s equation from (2.17) implies ˇ ˛ @t A C 2 PE D 0: R Q dv and There only remains to evaluate the flux terms 1 R3 Lg M R 1 Q Lg Mdv in (2.25) and (2.26). Following [10, 11], this is done by employing R3 (2.9) to evaluate that 1 ˛ Lg D Q.g ; g / v rx g C E v C O./;

(2.27)

30


which yields formally in the limit, by virtue of the infinitesimal Maxwellian form (2.11), h˛ i 1 E v lim Lg D Q.g; g/ v rx g C !0 h˛ i 1 D L g 2 v rx g C Ev 2 4 1 2 jvj 1 t D u L. /u C u L. / C L 2 2 4 h i 2 ˛ jvj u C u C C Ev div . C /v C 3 4 1 1 jvj D ut L. /u C u L. / C 2 L div . u C / ; 2 2 4

where we have used, in the last line, that div u D 0 and rx .C / D ˛ E, whatever the asymptotic regime. Next, we use that Q and Q have similar symmetry properties as and , thanks to the rotational invariance of L. More precisely, following [28], it can be shown (see also [14, Section 2.2.3]) that there exist two scalar valued functions ˛; ˇ W Œ0; 1/ ! R such that Q .v/ D ˛ .jvj/ .v/

and

Q .v/ D ˇ .jvj/ .v/;

which implies (see [11, Lemma 4.4]) that Z 2 ij Q kl M dv D ıi k ıj l C ıi l ıj k ıij ıkl ; 3 R3 Z 5 Q i j M dv D ıij ; 2 R3 where 1 D 10

Z R3

Q dv W M

and

2 D 15

Z

Q M dv:

R3

Hence, we conclude through tedious but straightforward calculations that Z Z Z 1 1 t Q dv Q dv D Lg M div . u/ M u u M dv lim !0 R3 R3 2 R3 juj2 Du˝u Id rx u C rxt u ; 3 Z Z Z 1 Lg Q M dv D u M dv div . / Q M dv lim !0 R3 R3 R3 5 5 D u rx : 2 2

(2.28)

(2.29)


31

We finally identify the advection and diffusion terms Z 1 Q dv D P .u rx u/ x u; Lg M lim P div !0 R3 Z 5 5 1 Lg Q M dv D u rx x : lim div !0 3 2 2 R On the whole, provided nonlinear terms remain stable in the limiting process, we obtain the following asymptotic systems: (1) When 1, letting tend to zero in the system (2.25) coupled with Maxwell’s equations (2.17) yields 8 h˛i ˇ ˆ ˆ u ^ B; @t u C u rx u x u D rx p C 2 E C ˆ ˆ ˆ ˆ ˆ < @ C u r D 0; t x x ˆ ˇ ˆ ˆ u; Œ @t E rot B D ˆ ˆ ˆ ˆ : Œ @t B C rot E D 0; with the constraints from (2.20) div u D 0; C D 0; div E D 0; div B D 0: (2) When D o.1/ and ˛ D O. 2 /, letting tend to zero in the system (2.25) coupled with Maxwell’s equations (2.17) yields @t u C u rx u x u D rx p; @t C u rx x D 0; with the constraints from (2.22) div u D 0; C D 0; E D 0; B D 0: (3) When D o.1/ and ˛2 is unbounded, letting tend to zero in the system (2.26) coupled with Maxwell’s equations (2.17) yields 8 h˛ i ˇ ˇ ˆ ˆ @t u C E C A C u rx u x u D rx p C u ^ B; < h˛ i 5 5 3 ˆ ˆ : @t C u rx x D u E; 2 2 2

32


with the constraints from (2.24) h˛ i div u D 0; rx . C / D E; ˇ u; rot E D 0; rot B D h˛ i div E D ; div B D 0; rot A D B; div A D 0: The above system can be rewritten more explicitly by defining the adjusted electric field EQ D @t A C E. It then holds that 8 ˇ Q ˆ ˆ E C rx @t u C u rx u x u D rx p C ˆ ˆ ˆ ˆ ˆ ˆ ˇ ˆ < u ^ B; C ˆ ˆ 5 3 3 ˆ ˆ ˆ @t C u rx x D 0; ˆ ˆ 2 2 2 ˆ ˆ : @t B C rot EQ D 0; with the constraints h ˛ i2 div u D 0;

x . C / D ; ˇ rot B D u; div B D 0; h˛ i div EQ D : Notice, finally, that if further ˛ D o./, then the above system is greatly simplified and becomes 8 ˇ Q ˇ ˆ ˆ ˆ < @t u C u rx u x u D rx p C E C u ^ B; @t C u rx x D 0; ˆ ˆ ˆ : @t B C rot EQ D 0; with the constraints div u D 0; C D 0; div EQ D 0; div B D 0; ˇ rot B D u; E D 0:


33

2.3.4 Summary At last, we see that the asymptotics of the Vlasov–Maxwell–Boltzmann system (2.7) can be depicted in terms of the limits of the following parameters: the strength of the electric induction ˛, the strength of the magnetic induction ˇ D

˛ ,

the ratio of the bulk velocity to the speed of light . Figure 2.1 summarizes the different asymptotic regimes, on a logarithmic scale, of the Vlasov–Maxwell–Boltzmann system (2.7). 1

ϵ

incompressible quasi-static Navier-Stokes-Fourier-Maxwell

ϵ2

γ incompressible Navier-Stokes-Fourier-Poisson

ϵ

incompressible Navier-Stokes-Fourier-Maxwell

incompressible quasi-static Navier-Stokes-Fourier-Maxwell-Poisson

ϵ2

incompressible Navier-Stokes-Fourier incompressible Navier-Stokes-Fourier ϵ3 α

Figure 2.1. Asymptotic regimes of the one-species Vlasov–Maxwell–Boltzmann system (2.7) on a logarithmic scale.

Thus, up to multiplicative constants, we reach the following asymptotic systems of equations: (1) If ˛ D o./ and ˇ D o./, we obtain the incompressible Navier–Stokes– Fourier system @t u C u rx u x u D rx p; div u D 0; (2.30) @t C u rx x D 0; C D 0:

34


This system satisfies the following formal energy conservation laws: 1 d kuk2L2 C krx uk2L2 D 0; x x 2 dt 1d k k2L2 C krx k2L2 D 0: x x 2 dt (2) If ˛ D 2 and D 1, we obtain the incompressible Navier–Stokes–Fourier– Maxwell system 8 @t u C u rx u x u D rx p C E C u ^ B; div u D 0; ˆ ˆ ˆ < @ C u r D 0; C D 0; t x x (2.31) ˆ div E D 0; @t E rot B D u; ˆ ˆ : @t B C rot E D 0; div B D 0: This system satisfies the following formal energy conservation laws:

1 d kuk2L2 C kEk2L2 C kBk2L2 C krx uk2L2 D 0; x x x x 2 dt 1 d k k2L2 C krx k2L2 D 0: x x 2 dt (3) If ˛ D o./, ˇ D and D o.1/, we obtain the incompressible quasi-static Navier–Stokes–Fourier–Maxwell system 8 @t u C u rx u x u D rx p C E C u ^ B; div u D 0; ˆ ˆ ˆ < @ C u r D 0; C D 0; t x x (2.32) ˆ rot B D u; div E D 0; ˆ ˆ : div B D 0: @t B C rot E D 0; This system satisfies the following formal energy conservation laws:

1d kuk2L2 C kBk2L2 C krx uk2L2 D 0; x x x 2 dt 1 d k k2L2 C krx k2L2 D 0: x x 2 dt Here, the electric field is defined indirectly as a mere distribution, through Faraday’s equation, by E D @t A; where B D rot A and div A D 0.


35

(4) If ˛ D and D , we obtain the incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system 8 @t u C u rx u x u D rx p C E C rx ˆ ˆ ˆ ˆ ˆ C u ^ B; div u D 0; ˆ ˆ ˆ < 3 5 3 @t

x . C / D ; C u rx x D 0; ˆ 2 2 2 ˆ ˆ ˆ ˆ rot B D u; div E D ; ˆ ˆ ˆ : div B D 0: @t B C rot E D 0; (2.33) This system satisfies the following formal energy conservation law (see Proposition 3.1 for an explicit computation of the energy): ? 2 3 1 d 2 2 2 2 kkL2 C kukL2 C k kL2 C P E L2 C kBkL2 x x x x x 2 dt 2 5 C krx uk2L2 C krx k2L2 D 0: x x 2 Here, the solenoidal component of the electric field is defined indirectly as a mere distribution, through Faraday’s equation, by PE D @t A; where B D rot A and div A D 0, while its irrotational component is determined, through Gauss’ law, by P ? E D rx . C / : Notice that the equations in this system are all coupled. (5) If ˛ D and D o./, we obtain the incompressible Navier–Stokes–Fourier– Poisson system 8 @t u C u rx u x u D rx p C rx ; ˆ ˆ ˆ < div u D 0; ˆ 5 3 3 ˆ ˆ

x . C / D : C u rx x D 0; : @t 2 2 2 (2.34) This system satisfies the following formal energy conservation law: 1d 3 2 2 2 2 C C C . C /k k kL2 krx kkL2 kukL2 L2 x x x x 2 dt 2 5 C krx uk2L2 C krx k2L2 D 0: x x 2

36


Physically, in this system, the fluid is subject to a self-induced static electric field E determined by rot E D 0; hence

div E D ;

E D rx . C /:

Notice that the equations in this system are all coupled.

2.3.5 The Vlasov–Poisson–Boltzmann system The Vlasov–Poisson–Boltzmann system describes the evolution of a gas of one species of charged particles (ions or electrons) subject to an self-induced electrostatic force. This system is obtained formally from the Vlasov–Maxwell–Boltzmann system by letting the speed of light tend to infinity while all other parameters remain fixed. Accordingly, setting D 0 in (2.7) yields the scaled Vlasov–Poisson–Boltzmann system 8 1 ˆ ˆ @t f C v rx f C ˛rx rv f D Q.f ; f /; ˆ ˆ < f D M .1 C g / ; (2.35) Z ˆ ˆ ˆ ˛ ˆ : g M dv:

x D R3 Here, the plasma is subject to a self-induced electrostatic field E determined by Z ˛ div E D g M dv; rot E D 0; R3 hence E D rx : The above system is supplemented with some initial data satisfying Z 1 in 1 H f jE in j2 dx < 1: C 2 2 R3 In particular, solutions of (2.35) satisfy the corresponding scaled entropy inequality, where t > 0, Z Z Z 1 t 1 1 2 H f C jE j dx C 4 D.f /.s/ dxds 2 2 R3 0 R3 Z 1 1 jE in j2 dx: 2 H fin C 2 R3 Thus, the formal asymptotic analysis of (2.35) is contained in our analysis of the Vlasov–Maxwell–Boltzmann system (2.7). Specifically, setting D ˇ D 0 in the limiting systems obtained in Section 2.3.3, we see that the Vlasov–Poisson– Boltzmann system (2.35) converges, when ˛ D o./, towards the incompressible Navier–Stokes–Fourier system in a Boussinesq regime, with E D 0: @t u C u rx u x u D rx p; div u D 0 @t C u rx x D 0; C D 0:

2.4 Formal analysis of the two-species asymptotics

37

In the case when ˛ ¤ 0, we find convergence towards the incompressible Navier– Stokes–Fourier–Poisson system: 8 @t u C u rx u x u D rx p ˆ ˆ < C rx ; div u D 0; h ˛ i2 ˆ 5 3 3 ˆ : @t

x . C / D ; C u rx x D 0; 2 2 2

where the electrostatic field is determined by ˛ E D rx . C /. In fact, the Vlasov–Poisson–Boltzmann system is inherently simpler than the Vlasov–Maxwell–Boltzmann system, because it couples the Vlasov–Boltzmann equation with a simple elliptic equation, namely Poisson’s equation, while the Vlasov– Maxwell–Boltzmann system couples the Vlasov–Boltzmann equation with an hyperbolic system, namely Maxwell’s system of equations. Thus, the rigorous mathematical analysis of the Vlasov–Maxwell–Boltzmann system, presented in the remainder of this work, will also apply to the Vlasov–Poisson–Boltzmann system and, therefore, analogous results will hold.

2.4 Formal analysis of the two-species asymptotics We turn now to the formal asymptotic study of the incompressible viscous regimes of the two-species Vlasov–Maxwell–Boltzmann system (1.2). Recall that we are only considering the case of equal masses and opposite charges. The analysis follows exactly the same steps as in the one-species case (1.1). However, the situation obviously becomes now more complex and general. There is indeed an additional parameter, measuring the strength of the interspecies interaction. A complete summary of all limiting systems corresponding to the weak and strong interspecies interactions is given in Section 2.4.6, below. The limiting systems corresponding to the simpler case of very weak interspecies interactions are listed in Section 2.4.2. For a plasma of two species of particles, our starting point is the scaled system 8 1 ı2 ˆ ˙ ˙ ˙ ˙ ˙ ˆ f Cv r f ˙.˛E Cˇv ^B / r f D ; f /C Q.f Q.f˙ ; f /; @ ˆ t x v ˆ ˆ ˆ ˆ ˆ ˆ f˙ D M 1Cg˙ ; ˆ ˆ Z ˆ ˆ C ˆ ˇ < g g vM dv; @t E rot B D R3 ˆ ˆ ˆ @t B Crot E D 0; ˆ ˆ Z ˆ ˆ C ˆ ˛ ˆ ˆ D g g M dv; div E ˆ ˆ R3 ˆ ˆ : div B D 0; (2.36)

38


supplemented with some initial data satisfying Z in 2 1 in 1 1 Cin H f H f C C jE j C jBin j2 dx < 1; 2 2 2 R3 where H.f˙in / D H.f˙in jM /. In particular, the corresponding scaled entropy inequality, where t > 0, Z 1 C 1 1 H f H f C C jE j2 C jB j2 dx 2 2 2 R3 Z tZ 1 (2.37) D fC C D f C ı2 D fC ; f .s/ dxds C 4 0 R3 Z 1 in 2 1 1 jE j C jBin j2 dx; 2 H fCin C 2 H fin C 2 R3 guarantees that the solution will remain – for all non-negative times – a fluctuation of order around the global equilibrium M , f˙ D M.1 C g˙ /: Note that the kinetic equations in (2.36) can then be rewritten, in terms of the fluctuation g , as C C C ˛ g g g 1 C gC C v rx C .˛E C ˇv ^ B / rv E v @t g g g 1 g C 1 Lg C ı2 L gC ; g Q.gC ; gC / C ı2 Q gC ; g D C ; Q.g ; g / C ı2 Q g ; gC Lg C ı2 L g ; gC (2.38) where we denote

L.g; h/ D

1 .Q.Mg; M / C Q.M; M h// M

and Q.g; h/ D

1 Q.Mg; M h/: M (2.39)

It turns out that, in the limit ! 0, we will have now three types of constraints: conditions on the velocity profiles coming from the fast relaxation towards thermodynamic equilibrium (i.e., small Knudsen regime, see Section 2.4.1); linear macroscopic hydrodynamic constraints due to the weak compressibility (i.e., small Mach regime, see Section 2.4.3); nonlinear macroscopic electrodynamic constraints coming from momentum and energy exchange between species due to interspecies collisions (see Section 2.4.5). As in the case of one species of charged particles, we expect the first two types of constraints to be weakly stable, and thus to be derived from simple uniform a priori estimates. The procedure leading to the last couple of electrodynamic constraint


39

equations (including Ohm’s law) is a little bit more complex and will depend on the strength of interspecies collisional interactions, that is to say, on the size of ı > 0 compared to > 0. In fact, the nature of the whole asymptotic systems obtained in the limit ! 0 will be conditioned by the size of ı, and we will therefore distinguish three different asymptotic regimes: Very weak interspecies collisional interactions, ı D O./; in this regime, the interspecies collision operators ı2 Q.f˙ ; f / in (2.36) are a regular perturbation. Therefore, the corresponding limiting systems will be composed of two hydrodynamic systems – one for each species – coupled mainly through the mean field interactions of the electromagnetic forces. The derivation of these regimes will be easily deduced from the asymptotic analysis for one species from Section 2.3 and will therefore be treated first in Section 2.4.2. Weak interspecies collisional interactions, ı D o.1/ and ı unbounded; in this regime, the interspecies collision operators ı2 Q.f˙ ; f / in (2.36) are a singular perturbation, whose order may vary from the other singular perturbations present in the system (2.36). In particular, it is not the most singular perturbation of (2.36). Strong interspecies collisional interactions, ı 1; in this regime, the interspecies collision operators ı2 Q.f˙ ; f / in (2.36) are a singular perturbation of the most singular order present in the system (2.36).

2.4.1 Thermodynamic equilibrium The entropy inequality (2.37) provides uniform bounds on E , B , gC , and g . Therefore, assuming some formal compactness, up to extraction of subsequences, one has E * E; B * B; g˙ * g ˙ ; in a weak sense to be rigorously detailed in a subsequent chapter. Then, multiplying (2.38) by and taking formal limits as ! 0 shows that C C 0 g g lim Lı D L D ; Œı 0 g g !0

40 where


Lg C ı2 L g; h g Lı D h Lh C ı2 L h; g ! R Lg C ı2 R3 S2 g C h g 0 h0 bM dv d R ; D Lh C ı2 R3 S2 h C g h0 g0 bM dv d

and LŒı

Lg C Œı2 Lg; h g D h Lh C Œı2 L h; g ! R Lg C Œı2 R3 S2 g C h g 0 h0 bM dv d : R D Lh C Œı2 R3 S2 h C g h0 g0 bM dv d

It can be shown (see Proposition 5.7) that, when Œı ¤ 0, the kernels of the vectorial linearized Boltzmann operators Lı and LŒı coincide exactly with the vector space spanned by the set 2 v v jvj 1 0 v : (2.40) ; ; 1 ; 2 ; 3 ; 0 1 v1 v2 v3 jvj2 However, when Œı D 0, the kernel of LŒı is larger and is composed of all vectors '1 .v/ '2 .v/ such that '1 .v/ and '2 .v/ are collision invariants whose coefficients are independent. Thus, we conclude, if Œı ¤ 0, that g ˙ is an infinitesimal Maxwellian of the form 0

1 2 C jvj 3 C C u v C g 22 2 A ; D@ (2.41) g C u v C jvj2 32 while, if Œı D 0,

C

g g

0 D@

C u v C

jvj2 2 jvj2 2

C C uC v C C

1

3 2 A ; 3 2

(2.42)

where C ; 2 R, u; uC ; u 2 R3 and ; C ; 2 R only depend on t and x, and are respectively the fluctuations of density, bulk velocity, and temperature. In fact, whenever ı is unbounded, we show below that necessarily uC D u and C D , as well, because of higher-order singular limiting constraints. Therefore, the infinitesimal Maxwellian form (2.42) will be assumed by the limiting fluctuations in the case ı D O./ only, that is, in the case of very weak interspecies collisions.


41

The fact that the fluctuations assume the infinitesimal Maxwellian form describes that the gas reaches thermodynamic (or statistical) equilibrium, in the fast relaxation limit. We define now the macroscopic fluctuations of density ˙ , bulk velocity u˙ and temperature ˙ by Z ˙ D

u˙ ˙

R3

Z

g˙ M dv;

g˙ vM dv; 2 Z jvj D g˙ 1 M dv; 3 R3 D

R3

and the hydrodynamic projection …g˙ of g˙ by ˙ …g˙ D ˙ C u˙ v C

jvj2 3 ; 2 2

which is nothing but the orthogonal projection of g˙ onto the kernel of L in L2 .Mdv/. Note that the previous step establishing the convergence of g˙ towards thermodynamic equilibrium yields, in fact, the uniform boundedness of C 1 g : Lı g Therefore, if Œı ¤ 0, we deduce, at least formally, that ! C C gC Cg C … g 2 2 C D O./; g C Cg g C… 2

where

C 2 C 2

2

gC Cg 2 gC Cg … 2

C… C

!

clearly defines the orthogonal projection of C g g onto the kernel of Lı , which is spanned by (2.40). This bound implies, in particular, that .gC C / .g / D O./

and

g˙ …g˙ D O./:

42


However, if Œı D 0, we can only formally deduce, for the moment, that g˙ …g˙ D O./: Just as in the one-species case (see Section 2.3.1), the convergence of gC and with a rate O./ towards their hydrodynamic projections …gC and …g can also be inferred, at from the uniform control of the entropy dissi least formally, pations 14 D fC and 14 D f in (2.37). We are now going to show how the exact same formal reasoning applied to the control of the mixed entropy dissipation ı2 D fC ; f in (2.37) yields formally that 4 ; (2.43) .gC C / .g / D O ı g

which is not so readily deduced by direct inspection of (2.38). Note that this control is relevant in the cases of weak or strong interspecies interactions only, that is when ı is unbounded. Thus, as in Section 2.3.1, formally approximating z log.1 C z/ by z 2 , which is correct in a neighborhood of z D 0, in the definition (1.16) of the mixed entropy dissipation, we deduce a control of !2 Z 0 0 ı2 fC f fC f fC f b dvdv d: 2 4 R3 R3 S2 fC f Then, since 0

0 f Cf fC f 0 0 D …gC C …g …gC …g C 0 C ! g …gC g …g 0 g …gC g …g 2 C C 0

0 ; gC g C 2 gC g 0 0 we infer that ı …gC C …g …gC …g is uniformly bounded. Finally, a direct computation of the integral Z h 0 C i2 ı2 C 0 C …g …g …g b dvdv d; …g 2 R3 R3 S2

shows that uC u D O

ı

and

C D O

ı

;

which incidentally establishes (2.43). Of course, the rigorous proof of such bounds, later on in Section 5.3.2, will necessitate the control of the large values of the fluctuations in order to justify the formal approximation of z log.1 C z/ by z 2 .


43

On the whole, we have shown that, for all cases of strong, weak and very weak interspecies interactions, it holds g˙ …g˙ D O./ and .gC C / .g / D O : (2.44) ı Note that this implies that ! C C gC Cg C … g 2 2 C gC Cg g C … 2 2 ! C C .g C C /.g / … 2 …g g D O : D C C C .g /.g / g …g ı … 2

We will therefore henceforth denote, when considering weak or strong interspecies collisions, " !# C C gC Cg ı C … gC h 2 2 : D C gC Cg h g C… 2

2

In particular, further note that, for weak interspecies collisions, that is, whenever ı D o.1/ and ı is unbounded, ! C C C C C / ı … .g /.g h …h h 2 D 0; (2.45) lim D lim C C …h !0 h !0 h … .g /.g / 2

h˙

˙

so that the weak limits * h are necessarily infinitesimal Maxwellians. But this does not seem to hold for strong interspecies interactions, that is, when ı 1. In light of the above formal controls, we define new macroscopic hydrodynamic variables C C uC C u C C ; u D ; D ; 2 2 2 and electrodynamic variables (irrelevant for very weak interspecies collisions because ı is bounded in this case) D

ı C ı C w D u u ; ; namely the electric charge n , the electric current j , and the internal electric energy w . We will also consider their formal weak limits n D C ;

* ;

u * u;

Notice that …h˙

j D

* ;

n * n;

j * j;

2 1 3 jvj j v C w ; D˙ 2 2 2

w * w:

44


hence, for weak interspecies collisions, lim

!0

h˙

2 1 3 jvj Dh D˙ j vCw ; 2 2 2 ˙

whereas, for strong interspecies collisions, we only have that 2 1 3 jvj ˙ lim …h˙ D …h D ˙ j v C w : !0 2 2 2

(2.46)

C The asymptotic dynamics of .C ; uC ; ; ; u ; /, or equivalently . ; u ; ; q ; j ; w /, is then governed by fluid equations, to be obtained from the moments equations associated with (2.38). Thus, successively multiplying (2.38) by the colli2 sion invariants 1, v and jvj2 , and integrating in Mdv, yields 8 ˆ ˆ @t ˙ C 1 div u˙ ˆ D 0; ˆ ˆ ˆ Z ˆ ˆ ˙ ˆ 1 ı2 ˛ ˆ ˙ ˆ @t u˙ r C C E C L g˙ ; g vM dv ˆ x 2 2 ˆ R3 ˆ ˆ ˆ ˆ ˆ ˇ ˛ ˆ ˙ ˙ ˆ ˆ < D ˙ E C u ^ B Z Z ı2 1 ˆ ˙ ˆ ˆ g M dv C Q g˙ ; g vM dv; div ˆ ˆ 3 3 R ˆ R ˆ Z ˆ ˆ ˙ jvj2 ˆ ı2 5 3 ˆ ˙ ˙ ˙ ˆ @t C C C L div u M dv g ; g ˆ ˆ 2 2 2 R3 2 ˆ ˆ ˆ Z Z ˆ ˆ jvj2 1 ı2 ˛ ˆ ˙ ˆ E g M dv C Q g˙ ; g div M dv; : D ˙ u˙ R3 2 R3 (2.47) where .v/ and .v/ have already been defined in (2.14). The above system will be used in the case of very weak interspecies interactions only. For weak and strong interspecies interactions, the evolution equations can then be recast, in terms of the new hydrodynamic and electrodynamic variables, as 8 1 ˆ ˆ @t C div u D 0; ˆ ˆ ˆ ˆ ˆ ˆ 1 ˇ ˛ ˆ ˆ u C . C / D E C ^ B @ r n j ˆ < t x 2 2ı Z (2.48) 1 gC C g ˆ ˆ ˆ div M dv; ˆ ˆ 2 ˆ R3 ˆ Z ˆ ˆ ˆ gC C g 1 ˛ 1 3 ˆ : @t C div u D j E div M dv: 2 2ı 2 R3


45

Recall that we are assuming ˛ D O./ and ˇ D O./ for very weak interspecies collisions, i.e., when ı D O./, and ˛ D O./ and ˇ D O.ı/ for weak and strong interspecies collisions, i.e., when ı is unbounded. Hence, the nonlinear terms in the right-hand side of (2.47), for very weak interspecies collisions, and (2.48), for weak and strong interspecies collisions, containing the electromagnetic fields, are expected to be bounded. Furthermore, just as in the case of one species, the polynomials .v/ 2 and .v/ are orthogonal to the collision R invariants in the L .Mdv/ inner product. R That is to say, R3 ' M dv D 0 and R3 ' M dv D 0, for all collision invariants '.v/. Since, according to (2.44), the fluctuations gC and g converge towards infinitesimal Maxwellians with a rate O./, it is therefore natural to expect, at least formally, that the terms Z Z ˙ 1 1 ˙ g M dv D g …g˙ M dv; R3 R3 Z Z ˙ 1 1 ˙ g M dv D g …g˙ M dv; R3 R3 in (2.47) and (2.48) are bounded and have a limit. Thus, following the strategy for one species in Section 2.3.1, we rewrite (2.47) as 8 ˆ ˆ @t ˙ C 1 div u˙ ˆ D 0; ˆ ˆ ˆ Z ˆ ˆ ˆ 1 ˙ ı2 ˛ ˆ ˙ ˆ @t u˙ r C C E C L g˙ ; g vM dv ˆ x 2 2 ˆ R3 ˆ ˆ ˆ ˆ ˆ ˇ ˛ ˆ ˙ ˙ ˆ ˆ < D ˙ E C u ^ B Z Z 1 ı2 ˆ ˙Q ˆ ˆ Lg M dv C Q g˙ ; g vM dv; div ˆ ˆ 3 3 R ˆ R ˆ ˆ 2 Z ˆ 2 ˆ 1 ı 3 ˆ ˙ ˙ ˙ jvj ˆ C C L ; g @ div u M dv g t ˆ ˆ 2 2 R3 2 ˆ ˆ ˆ Z Z ˆ ˆ jvj2 1 ˛ ˙ ı2 ˆ ˙ Q ˆ E L g Q g˙ ; g u div M dv; D ˙ M dv C : R3 2 R3 (2.49) and (2.48) as 8 1 ˆ ˆ @t C div u D 0; ˆ ˆ ˆ ˆ ˆ ˆ 1 ˇ ˛ ˆ ˆ ˆ u C . C / D E C ^ B @ r n j < t x 2 2ı Z (2.50) C 1 ˆ Q ˆ ˆ L g C g M dv; div ˆ ˆ 2 ˆ R3 ˆ Z ˆ ˆ 1 ˛ 1 3 ˆ ˆ C D E L gC C g Q M dv; @ div u j div : t 2 2ı 2 R3 and

46


where Q and Q are the pseudo-inverses of and , respectively, defined in (2.15), and where the terms 1 Lg˙ will be expressed employing the Vlasov–Boltzmann equations (2.38). Each of the above macroscopic systems (2.49) and (2.50) is coupled with Maxwell’s equations on E and B : 8 ˇ ˇ ˆ ˆ @t E rot B D uC ˆ u D j ; ˆ ı ˆ ˆ < @t B C rot E D 0; (2.51) ˛ C ˛ ˆ ˆ n div E D ; D ˆ ˆ ˆ ˆ : div B D 0: A careful formal analysis of the whole coupled macroscopic systems (2.49)– (2.51), for very weak interspecies collisions, and (2.50)–(2.51), for weak and strong interspecies collisions, will yield the asymptotic dynamics of .˙ ; u˙ ; ˙ ; E; B/ and .; u; ; E; B/, respectively. However, note that, in the case of weak or strong interspecies collisions only, the above coupled system (2.50)–(2.51) remains underdetermined, as the evolution for n , j and w is missing. It turns out that the electrodynamic variables will be determined by nonlinear constraint equations. In particular, j will be asymptotically determined by the so-called Ohm’s law, which we derive below in Section 2.4.5.

2.4.2 The case of very weak interspecies collisions The reader should, at this point, take some time to compare the two-species system (2.49)–(2.51) with the one-species system (2.16)–(2.17). When ı D O./, the coupling between cations and anions in the two-species system (2.49)–(2.51) is caused only by the mean field interaction of the electromagnetic field .E; B/ and by the low order interspecies collision terms Z Z ˙ v ˙ v ı2 ı2 and L g ; g Q g ; g jvj2 M dv jvj2 M dv: 2 R3 R3 2 2 As we are about to discuss, the system (2.49)–(2.51) essentially behaves, in the limit ! 0, as two coupled one-species systems of the kind (2.16)–(2.17). Indeed, when compared with (2.16), the only additional terms that one finds in (2.49) are: The linear interspecies collision terms Z ˙ v ı2 L g ; g jvj2 M dv; 2 R3 2 which converge, as ! 0, towards 2 1 C 2 Z ˙ v ı ı u u C ; L g ;g 5 1 jvj2 M dv D ˙

R3 2 2


47

where the electrical conductivity > 0 and the energy conductivity > 0 are constants defined by Z 1 1 v L .v; v/ M dv D 2 R3 Z ˇ ˇ 1 ˇv v 0 ˇ2 b.v v ; /MM dvdv d D 2 R3 R3 S2 Z 1 D jv v j2 m.v v /MM dvdv ; 2 R3 R3 and

Z 1 1 jvj2 L jvj2 ; jvj2 M dv D

20 R3 Z 2 2 1 jvj jv 0 j2 b.v v ; /MM dvdv d D 20 R3 R3 S2 Z 2 2 1 jvj jv j2 m.v v /MM dvdv ; D 20 R3 R3

where the cross-section for momentum and energy transfer m.vv / is defined in Proposition A.1. The nonlinear interspecies collisions terms Z v ı2 Q g˙ ; g jvj2 M dv; R3 2 which are at least of formal order O./, and hence vanish in the limit ! 0. Thus, the remainder of the formal asymptotic analysis of the two-species system (2.49)–(2.51) follows exactly the same steps as the analysis of the one-species system (2.16)–(2.17) performed in Sections 2.3.2 and 2.3.3, which we somewhat detail now. Note first that the system (2.49)–(2.51) can be rewritten as a singular perturbation 0

1 0 C 1 C C Bqu C BquC C B C B C B 3 CC B 3 CC B 2 C B 2 C B C B C B C B C @t B C C W B C D O.1/; B qu C B qu C B C B C 3 B C B 3 C B 2 C B 2 C @ E A @ E A B B

48


which describes the propagation of waves in the system, where the wave operator is given by 1 0 1 0 div 0 0 0 0 0 0 q C B1 1 2 ˛ C B rx 0 r 0 0 0 0 x 2 Id 3 C B q C B 1 2 C B 0 div 0 0 0 0 0 0 3 C B C B 0 1 div 0 0 0 0 0 0 C B q W D B C: 1 1 2 ˛ B 0 r 0 rx 0 C 0 0 2 Id x 3 C B q C B 1 2 B 0 div 0 0 0 C 0 0 0 C B 3 C B ˛ ˛ 1 Id 0 0 Id 0 0 rot A @ 0 2 2 1 rot 0 0 0 0 0 0 0 We derive then the macroscopic constraint equations on ˙ ; u˙ ; ˙ ; E; B reproducing the reasoning from Section 2.3.2. (1) When 1 (so that ˛ D O. 2 /), averaging over fast time-oscillations as ! 0, we get the macroscopic constraints div u˙ D 0;

˙ C ˙ D 0;

respectively referred to as incompressibility and Boussinesq relations. These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.51), div E D 0; div B D 0: (2) When D o.1/ and ˛ D O. 2 /, averaging over fast time-oscillations as ! 0, we get the macroscopic constraints div u˙ D 0; ˙ C ˙ D 0; rot B D 0; rot E D 0: These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.51), div E D 0; div B D 0: Hence,

E D 0;

B D 0:

(3) When D o.1/ and ˛2 is unbounded, averaging over fast time-oscillations as ! 0, we get the macroscopic constraints h˛ i rx ˙ C ˙ D ˙ E; div u˙ D 0; ˇ C rot E D 0: u u ; rot B D


49

As usual, when ˛ D o./, the weak Boussinesq relation rx .˙ C ˙ / D 0 can be improved to the strong Boussinesq relation ˙ C ˙ D 0, assuming ˙ and ˙ enjoy enough integrability. These are supplemented by the asymptotic constraints coming from Gauss’ laws in (2.51), h˛ i div E D div B D 0: C ; Next, following the reasoning from Section 2.3.3, we derive the asymptotic evolution equations associated with the two-species system (2.49)–(2.51). To this end, notice that (2.38) implies, in particular, that ˛ 1 ˙ Lg D Q.g˙ ; g˙ / v rx g˙ ˙ E v C O./; which is analogous to (2.27) in the one-species case. Hence, we obtain the advection and diffusion terms, as in Section 2.3.3: Z 1 Q dv D P u˙ rx u˙ x u˙ ; lim P div Lg˙ M !0 3 ZR 5 5 1 Lg˙ Q M dv D u˙ rx ˙ x ˙ ; lim div !0 3 2 2 R where D

1 10

Z R3

Q dv W M

and

D

2 15

Z

Q M dv:

R3

We are now in a position to obtain the limiting evolution of the system (2.49)– (2.51). (1) When 1, letting tend to zero in the system (2.49)–(2.51) yields 8 2 1 C ı ˆ ˙ ˙ ˙ ˙ ˆ ˆ u u @t u C u rx u x u ˙ ˆ ˆ ˆ ˆ ˆ h˛i ˆ ˇ ˆ ˆ D rx p˙ ˙ 2 E ˙ u˙ ^ B; ˆ ˆ ˆ < 2 ı 1 C ˙ ˙ ˙ ˙ ˆ C u r ˙ D 0; @ ˆ t x x ˆ

ˆ ˆ ˆ ˆ ˆ ˇ ˆ C ˆ Œ @ E rot B D u ; u t ˆ ˆ ˆ ˆ : Œ @t B C rot E D 0;

50


with the constraints div u˙ D 0; ˙ C ˙ D 0; div E D 0; div B D 0: (2) When D o.1/ and ˛ D O. 2 /, letting tend to zero in the system (2.49)– (2.51) yields 8 2 ˆ 1 C ı ˙ ˙ ˙ ˙ ˆ ˆ u C u r u u ˙ u u D rx p˙ ; @ x x < t 2 ˆ ı 1 C ˆ ˆ D 0; : @t ˙ C u˙ rx ˙ x ˙ ˙

with the constraints div u˙ D 0; ˙ C ˙ D 0; E D 0; B D 0: (3) When D o.1/ and ˛2 is unbounded, letting tend to zero in the system (2.49)–(2.51) yields 8 2 ˆ 1 C ˇ ı ˙ ˙ ˙ ˙ ˆ ˆ u ˙ r u u ˙ u A C u u @ t x x ˆ ˆ ˆ ˆ ˆ h˛ i ˆ ˇ ˆ ˆ < D rx p˙ ˙ ˙ E ˙ u˙ ^ B; ˆ ˆ 5 ˙ 5 5 ı 21 C 3 ˙ ˆ ˙ ˙ ˙ ˆ C u rx x ˙ @t ˆ ˆ ˆ 2 2 2 2

ˆ ˆ h i ˆ ˆ : D ˙ ˛ u˙ E; with the constraints

h˛ i E; div u˙ D 0; rx ˙ C ˙ D ˙ ˇ C u u ; rot B D rot E D 0; h˛ i div E D C ; div B D 0; rot A D B; div A D 0:


51

The above system can be rewritten more explicitly by defining the adjusted electric field EQ D @t A C E. It then holds that 8 2 1 C ı ˆ ˙ ˙ ˙ ˙ ˆ u C u r u u ˙ u u @ ˆ t x x ˆ ˆ ˆ ˆ ˆ ˆ ˇ Q ˇ ˙ ˆ ˙ ˙ ˙ ˆ D r p ˙ r ˙ u ^ B; E C ˆ x x ˆ ˆ < 5 3 ˙ 3 ˙ ˙ ˙ ˙ C u x ˙ r @ ˆ t x ˆ ˆ 2 2 2 ˆ ˆ 2 ˆ ˆ 5 ı 1 C ˆ ˆ ˆ ˙ D 0; ˆ ˆ 2

ˆ ˆ : @t B C rot EQ D 0; with the constraints

h ˛ i2

x ˙ C ˙ D ˙ C ; div u˙ D 0; ˇ C div B D 0; rot B D u u ; h˛ i div EQ D C :

Notice, finally, that if further ˛ D o./, then the above system is greatly simplified and becomes 8 2 ˆ 1 C ı ˙ ˙ ˙ ˙ ˆ ˆ u C u r u u ˙ u u @ t x x ˆ ˆ ˆ ˆ ˆ ˆ ˆ < D r p˙ ˙ ˇ EQ ˙ ˇ u˙ ^ B; x ˆ 2 ˆ ˆ ı 1 C ˆ ˙ ˙ ˙ ˙ ˆ D 0; @t C u rx x ˙ ˆ ˆ ˆ

ˆ ˆ : Q @t B C rot E D 0; with the constraints ˙ C ˙ D 0; div u˙ D 0; div EQ D 0; div B D 0; ˇ C rot B D E D 0: u u ;

52


On the whole, we conclude that, in the case of very weak interspecies collisions ı D O./, the parameters ˛, ˇ and determine the asymptotics of the two-species Vlasov–Maxwell–Boltzmann system (2.36) exactly as they do determine the asymptotics of the one-species Vlasov–Maxwell–Boltzmann system treated in Section 2.3. More precisely, the limiting two-fluid macroscopic systems we obtain here can always be interpreted as two systems for one species – one for cations and one for anions – coupled through their mean field interaction with the electromagnetic field .E; B/ and, whenever ı , by an interspecies exchange of momentum and energy expressed by the linear terms 1 .uC u / and 1 . C /. Therefore, the different asymptotic regimes for two species are also described by Figure 2.1 on page 33. Thus, when ı , up to multiplicative constants, we arrive at the following asymptotic systems of equations: (1) If ˛ D o./ and ˇ D o./, we obtain the two-fluid incompressible Navier– Stokes–Fourier system 8 1 C ˆ < @t u˙ C u˙ rx u˙ x u˙ ˙ u u D rx p˙ ; div u˙ D 0; ˆ : @ ˙ C u˙ r ˙ ˙ ˙ 1 C D 0; ˙ C ˙ D 0: t x x

(2.52) This system satisfies the following formal energy conservation laws:

1 d uC 2 2 C ku k2 2 C rx uC 2 2 C krx u k2 2 Lx Lx Lx Lx 2 dt 1 2 C uC u L2 D 0; x

1 d 2 C 2 C k k2 2 C rx C 2 2 C krx k2 2 Lx Lx Lx Lx 2 dt 1 2 C C L2 D 0: x

(2) If ˛ D 2 and D 1, we obtain the two-fluid incompressible Navier–Stokes– Fourier–Maxwell system 8 @t u˙ C u˙ rx u˙ x u˙ ˆ ˆ ˆ ˆ ˆ 1 C ˆ ˆ div u˙ D 0; u u D rx p˙ ˙ E ˙ u˙ ^ B; ˆ ˙ ˆ ˆ ˆ ˆ < @ ˙ C u˙ r ˙ ˙ t x x (2.53) ˆ 1 ˆ C ˙ ˙ ˆ ˙ C D 0; D 0; ˆ ˆ

ˆ ˆ ˆ ˆ ˆ @t E rot B D uC u ; div E D 0; ˆ ˆ : div B D 0: @t B C rot E D 0;


53

This system satisfies the following formal energy conservation laws:

1 d uC 2 2 C ku k2 2 C kEk2 2 C kBk2 2 Lx Lx Lx Lx 2 dt

1 2 2 C rx uC L2 C krx u k2L2 C uC u L2 D 0; x x x

1 d C 2 L2 C k k2L2 x x 2 dt

1 2 2 C rx C L2 C krx k2L2 C C L2 D 0: x x x

(3) If ˛ D o./, ˇ D and D o.1/, we obtain the two-fluid incompressible quasi-static Navier–Stokes–Fourier–Maxwell system 8 @t u˙ C u˙ rx u˙ x u˙ ˆ ˆ ˆ ˆ ˆ 1 C ˆ ˆ ˙ div u˙ D 0; u u D rx p˙ ˙ E ˙ u˙ ^ B; ˆ ˆ ˆ ˆ ˆ < @ ˙ C u˙ r ˙ ˙ t x x (2.54) ˆ 1 C ˆ ˙ ˙ ˆ ˙ C D 0; D 0; ˆ ˆ

ˆ ˆ ˆ ˆ ˆ div E D 0; rot B D uC u ; ˆ ˆ : @t B C rot E D 0; div B D 0: This system satisfies the following formal energy conservation laws:

1 d uC 2 2 C ku k2 2 C kBk2 2 Lx Lx Lx 2 dt

1 2 2 C rx uC L2 C krx u k2L2 C uC u L2 D 0; x x x

1 d C 2 2 C k k2 2 Lx Lx 2 dt

1 2 2 C rx C L2 C krx k2L2 C C L2 D 0: x x x

Here, the electric field is defined indirectly as a mere distribution, through Faraday’s equation, by E D @t A; where B D rot A and div A D 0.

54


(4) If ˛ D and D , we obtain the two-fluid incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system: 8 1 C ˆ ˆ u u @t u˙ C u˙ rx u˙ x u˙ ˙ ˆ ˆ ˆ ˆ ˆ ˙ ˙ ˙ ˆ p ˙ E C r ˙ u˙ ^ B; D r ˆ x x ˆ ˆ ˆ ˆ ˆ div u˙ D 0; ˆ ˆ ˆ ˆ ˆ 5 3 ˙ 3 ˙ ˆ ˙ ˙ ˙ ˆ C u x ˙ r @ t x ˆ ˆ 2 2 2 ˆ < 5 ˙ C D 0; ˆ ˆ 2

ˆ ˆ ˆ ˆ ˆ x ˙ C ˙ D ˙ C ; ˆ ˆ ˆ ˆ ˆ rot B D uC u ; ˆ ˆ ˆ ˆ ˆ @t B C rot E D 0; ˆ ˆ ˆ ˆ div E D C ; ˆ ˆ ˆ : div B D 0:

(2.55)

This system satisfies the following formal energy conservation law: 1 d 2 dt

C 2 2 C k k2 2 C uC 2 2 C ku k2 2 L L L L x

x

x

x

? 2 3 3 2 2 C 2 C L2 C k kL2 C P E L2 C kBkL2 x x x x 2 2

5 2 2 C rx uC L2 C krx u k2L2 C rx C L2 C krx k2L2 x x x x 2 1 5 2 2 C 2 D 0: C uC u L2 C Lx x 2

Here, the solenoidal component of the electric field is defined indirectly as a mere distribution, through Faraday’s equation, by PE D @t A; where B D rot A and div A D 0, while its irrotational component is determined, through Gauss’ law, by P ? E D ˙rx ˙ C ˙ : Notice that the equations in this system are all coupled.


55

(5) If ˛ D and D o./, we obtain the two-fluid incompressible Navier–Stokes– Fourier–Poisson system: 8 1 C ˆ u u D rx p˙ C ˙ rx ˙ ; @t u˙ C u˙ rx u˙ x u˙ ˙ ˆ ˆ ˆ ˆ ˆ ˆ div u˙ D 0; ˆ ˆ ˆ ˆ < 3 5 3 ˙ ˙ ˙ ˙ ˙ C u rx x ˙ @t ˆ 2 2 2 ˆ ˆ ˆ ˆ 5 C ˆ ˆ ˙ D 0; ˆ ˆ ˆ ˆ 2

:

x ˙ C ˙ D ˙ C : (2.56) This system satisfies the following formal energy conservation law: 1d C 2 2 C k k2 2 C uC 2 2 C ku k2 2 L Lx L Lx x x 2 dt 2 3 2 3 C C L2 C k k2L2 C rx ˙ C ˙ L2 x x x 2 2

2 5 2 C rx uC L2 C krx u k2L2 C rx C L2 C krx k2L2 x x x x 2 1 5 2 2 C 2 D 0: C uC u L2 C Lx x 2

Physically, in this system, the fluid is subject to a self-induced static electric field E determined by rot E D 0; hence

div E D C ;

E D ˙rx ˙ C ˙ :

Notice that the equations in this system are all coupled. When ı D o./, one obtains the corresponding asymptotic systems by simply discarding the linear terms ˙ 1 uC u and ˙ 1 C in the preceding systems. The above interpretation of two-fluid systems as a coupling of one-fluid systems will no longer hold for the more singular case of weak and strong interactions, i.e., when ı is unbounded, which we treat next.

2.4.3 Macroscopic hydrodynamic constraints Let us focus now on the analysis of the weak and strong interspecies collisional interactions. Contrary to the one-species case, in the two-species case, when ı is unbounded, the acoustic waves are always decoupled from the electromagnetic waves, which we treat below in Section 2.4.5. We deal now with the acoustic waves.

56


At leading order, the system of equations (2.50) describes the propagation of acoustic ! r 3 ; u ; 2 waves:

0

1 0 1 B C B C @t @qu A C W @qu A D O.1/; 3 3 2 2

(2.57)

where the wave operator W , containing the singular terms from (2.50), is defined by 1 0 1 div 0 q0 C B1 1 2 rx 0 rx C : W D B (2.58) 3 A @ q 0

1

2 3

div

0

The wave operator W is antisymmetric (with respect to the L2 .dx/ inner product), and so can only have purely imaginary eigenvalues. The semi-group generated by this operator may thus produce fast time-oscillations. Consequently, averaging over fast time-oscillations as ! 0, we get the macroscopic constraints div u D 0;

C D 0;

(2.59)

respectively referred to as incompressibility and Boussinesq relations. The exact nature of time oscillations produced by the system (2.57), in the limit ! 0, will be rigorously discussed, with greater detail, later on in Chapter 10.

2.4.4 Hydrodynamic evolution equations The previous step shows that, since W is singular, the asymptotic dynamics of ! r 3 ; u ; 2 becomes constrained to the kernel Ker W as ! 0. Moreover, since W is antisymmetric, its range is necessarily orthogonal to its kernel. Therefore, in order to get the asymptotic evolution equations for r ! 3 ; u; ; 2 it is natural to project the system (2.57) onto Ker W , which will rid us of all the singular terms in (2.57) and allow us to pass to the limit. In other words, we will


57

obtain the limiting dynamics of the system (2.57) by testing it against functions in Ker W . The kernel of W , defined in (2.58), is obviously determined by all ! r 3 0 0 0 ; u ; 2 which satisfy

div u0 D 0

0 C 0 D 0:

and

It is then readily seen that its orthogonal complement Ker W? is determined by all ! r 3Q Q ; uQ ; 2 such that P uQ D 0

and

3Q Q D 0: 2

Hence, projecting the system (2.50) onto Ker W? yields 8 C Z g C g Q 1 ˇ ˛ ˆ ˆ ˆ P div n j P u C L E C ^ B M dv D P @ ; < t 2 2 2ı R3 C Z ˆ 1 ˛ 3 g C g Q ˆ ˆ L C div M dv D j E : : @t 2 2 2ı R3 (2.60) There only remains to evaluate the flux terms C Z g C g Q 1 L M dv R3 2 and 1

Z

g C C g L 2 R3

Q M dv

in (2.60). Just as for one species in Section 2.3.3, following [10, 11], this is done by employing (2.38) to evaluate that

1 C ı2 C L g C g D Q.gC ; gC / C Q.g; g /

Cı2 Q gC ; g C Q g ; gC v rx gC Cg CO./;

58


which yields formally in the limit, by virtue of the infinitesimal Maxwellian form (2.41), C

1 1 g C g lim L Q.g C ; g C / C Q.g ; g / D 2 !0 2 2.1 C Œı / C

Œı2 Q g ; g C Q g; gC C 2 2.1 C Œı / C 1 g C g v rx 1 C Œı2 2 C 2 i 1 1 h C2 g C g Lg v r CL g D x 4 1 C Œı2 2 4 1 1 jvj D ut L. /u C u L. / C 2 L 2 2 4 2 jvj 1 div . C /v C u C u C 1 C Œı2 3 4 1 2 1 t jvj D u L. /u C u L. / C L 2 2 4 1 div . u C / ; 1 C Œı2 where we have used, in the last line, that div u D 0 and rx . C / D 0. Next, we use that Q and Q have similar symmetry properties as and , thanks to the rotational invariance of L. More precisely, following [28], it can be shown (see also [14, Section 2.2.3]) that there exist two scalar-valued functions ˛; ˇ W Œ0; 1/ ! R such that Q .v/ D ˛ .jvj/ .v/

and

Q .v/ D ˇ .jvj/ .v/;

which implies (see [11, Lemma 4.4]) that Z 2 ij Qkl M dv D .1 C Œı2 / ıi k ıj l C ıi l ıj k ıij ıkl ; 3 3 ZR 5 2 Q i j M dv D .1 C Œı / ıij ; 3 2 R where 1 D 10.1 C Œı2 /

Z

Q dv W M R3

and

2 D 15.1 C Œı2 /

Z R3

Q M dv: (2.61)

59


Hence, we conclude through tedious but straightforward calculations that C Z Z 1 1 t g C g Q lim L M dv D u u M dv !0 R3 2 R3 2 Z 1 Q dv div . u/ M 1 C Œı2 R3 juj2 Du˝u Id rx u C rxt u ; 3 C Z Z 1 g C g Q L u M dv M dv D lim !0 R3 2 R3 Z 1 div . / Q M dv 1 C Œı2 R3 5 5 D u rx : 2 2 We finally identify the advection and diffusion terms C Z 1 g C g Q lim P div L M dv D P .u rx u/ x u; !0 2 R3 and C Z 5 5 1 g C g Q lim div L M dv D u rx x : !0 2 2 2 R3 On the whole, provided nonlinear terms remain stable in the limiting process, we obtain the asymptotic system 8 h i < @ u C u r u u D r p C 1 ˛ nE C 1 ˇ j ^ B; t x x x 2 2 ı (2.62) : @t C u rx x D 0; with the constraints from (2.59) div u D 0;

C D 0:

(2.63)

Unfortunately, as will be discussed later on in Section 3.2, the rigorous weak stability of the nonlinear terms n E * nE and j ^ B * j ^ B remains unclear in general. This will be, in fact, one of the main reasons for the breakdown of the weak compactness method in the most singular cases of hydrodynamic limits of the twospecies Vlasov–Maxwell–Boltzmann system (2.36), which will lead us to develop new relative entropy methods and consider dissipative solutions (see Section 3.2.3 and Chapter 12). There only remains now to formally establish the asymptotic system for the electrodynamic variables .n; j; w/ and the electromagnetic field .E; B/, which we do next.

60


2.4.5 Macroscopic electrodynamic constraints and evolution The constraint equations for the electrodynamic variables will be obtained from the analysis of the difference of both components of (2.38): 1 C @t g g C v rx gC g ı ı ˛ ˇ ˛ C E C v ^ B rv gC C g E v 2 C gC C g ı ı ı 1 C C D 2 L hC h L h h ; h h ı 1 C ı2 1 ı2 C Q g C g ; n C Q n ; gC C g C 2ı 2ı 1 C ı2 C 1 ı2 C C Q g C g ; hC Q h h C h C ; g C g : 2 2 2ı 2ı (2.64) However, the analysis in the case ı 1 will slightly differ from the case ı D o.1/, with ı unbounded. We begin with the case ı 1 of strong interspecies interactions. First, integrating (2.64) in Mdv and letting ! 0 easily yields the continuity equation 1 @t n C div j D 0: (2.65) Œı Moreover, the above equation (2.64) contains no singular term in this situation. Therefore, letting ! 0 yields, employing (2.41), h˛i 1 ˇ Ev v rx n 2 .u ^ B/ v 2 Œı ı ı 1 D 2 L hC h L hC h ; h hC Œı jvj2 jvj2 ; u v : C ŒınL u v C 2 2 Further, introducing the linear operator Lg D L.g; g/;

(2.66)

we have

h˛i 1 ˇ rx n 2 .u ^ B/ 2 E v Œı ı ı 1 jvj2 2 L hC h L hC h : D ŒınL u v C 2 Œı Now, it can be shown that, in general, the linear operator 1 LCL Œı2

(2.67)


61

is self-adjoint and Fredholm of index zero on L2 .Mdv/ (or a variant of it depending on the cross-section, see Propositions 5.4 and 5.8). Therefore, its range is exactly the orthogonal complement of its kernel, which is composed of all constant functions (see Proposition 5.9). It follows that ˆ.v/ D v 2 L2 .Mdv/ and ‰.v/ D

jvj2 3 2 L2 .Mdv/ 2 2

Q 2 L2 .Mdv/ and belong to the range of Œı12 L C L and, thus, that there are inverses ˆ Q 2 L2 .Mdv/ such that ‰ ˆD

1 Q C Lˆ Q Lˆ Œı2

and

‰D

1 Q C L‰; Q L‰ Œı2

(2.68)

which can be uniquely determined by the fact that they are orthogonal to the kernel of 1 LCL Œı2 Q and ‰ Q have similar (i.e., to constant functions). Furthermore, it can be shown that ˆ symmetry properties as ˆ and ‰, thanks to the rotational invariance of L and L. More precisely, employing methods from [28] (see also [14, Section 2.2.3]), one verifies that there exist two scalar-valued functions ˛; ˇ W Œ0; 1/ ! R such that Q ˆ.v/ D ˛ .jvj/ ˆ.v/ which implies that

and

Q ‰.v/ D ˇ .jvj/ ‰.v/;

Z

Q j M dv D 1 ıij ; î ˆ 2 R3

(2.69)

Z 2 Q dv D ˆ ˆM (2.70) 3 R3 defines the electrical conductivity > 0. For completeness, we also define the energy conductivity > 0 by Z Q dv:

D ‰ ‰M (2.71) where

R3

Q integrating in Mdv, exploiting the self-adjointness Then, multiplying (2.67) by ˆ, of L and L and the limiting representation (2.46) of …h˙ , yields Ohm’s law h˛i 1 ˇ j Œınu D rx n C EC u^B : (2.72) 2Œı ı ı Q we obtain the energy equivalence relation Similarly, multiplying (2.67) by ‰, w D Œın:

(2.73)

Finally, in the case ı 1, the whole asymptotic system (2.62)-(2.63)-(2.65)(2.72)-(2.73) will be fully determined when considering the coupling with the

62


limiting Maxwell’s equations from (2.51): 8 ˇ ˆ ˆ j; Œ @t E rot B D ˆ ˆ ı ˆ ˆ < Œ @t B C rot E D 0; h˛ i ˆ ˆ ˆ div E D n; ˆ ˆ ˆ : div B D 0: Let us focus now on the case ı D o.1/, which turns out to be more complicated than the case ı 1, for it contains yet another singular limit, as we are about to h see. Indeed, the most singular term in (2.64) being ı12 L hC , we begin by projecting (2.64) onto the collision invariants in order to eliminate this singular term. This yields the system 8 1 ˆ ˆ @t n C div j D 0; ˆ ˆ ı ˆ ˆ Z ˆ 2 ˆ 1 2˛ ˆ C ˆ n @ j C C C L hC r w E ˆ t x h ; h h vM dv ˆ 2 ˆ ı ı ı ı R3 ˆ ˆ ˆ ˆ 2ˇ 2˛ ˆ ˆ D E C u ^ B ˆ ˆ ˆ ı ı ˆ ˆ Z ˆ ˆ 1 C ˆ ˆ div g …gC g C …g M dv ˆ ˆ ˆ ı R3 ˆ ˆ Z < C

Q g ; g Q g ; gC vM dv; Cı ˆ R3 ˆ ˆ 2 Z ˆ ˆ C 2 3 ˆ C jvj ˆ ˆ @ w C div j C L h ; h h h M dv t ˆ ˆ 2ı2 ı2 2 ˆ R3 ˆ ˆ ˆ 2˛ ˆ ˆ u E D ˆ ˆ ˆ ı ˆ Z ˆ ˆ 1 C ˆ ˆ ˆ div g …gC g C …g M dv ˆ ˆ ı ˆ R3 ˆ Z ˆ ˆ jvj2 ˆ ˆ : Q gC ; g Q g ; gC Cı M dv: 2 R3 (2.74) Remark. Observe that the system (2.74) may also be deduced directly from (2.47) by considering the difference of the equations for cations and anions. Then, since 2 1 3 jvj …h˙ D ˙ v C w j ; 2 2 2 straightforward computations based on symmetry of integrands show that Z 2 C L … hC vM dv D j h ; … h h 3 R Z 2 C jvj 1 C L … h h M dv D w ; … h h 2

R3


63

where the electrical conductivity > 0 and the energy conductivity > 0 are constants defined by Z 1 1 v L .v; v/ M dv D 6 R3 Z ˇ ˇ 1 ˇv v 0 ˇ2 b.v v ; /MM dvdv d D (2.75) 6 R3 R3 S2 Z 1 D jv v j2 m.v v /MM dvdv ; 6 R3 R3 and Z 1 1 jvj2 L jvj2 ; jvj2 M dv D

4 R3 Z 2 2 1 jvj jv 0 j2 b.v v ; /MM dvdv d D (2.76) 4 R3 R3 S2 Z 2 2 1 jvj jv j2 m.v v /MM dvdv ; D 4 R3 R3 where the cross-section for momentum and energy transfer m.v v / is defined in Proposition A.1. It follows that the system (2.74) may be rewritten as 8 1 ˆ ˆ @t n C div j D 0; ˆ ˆ ˆ ı ˆ ˆ

ˆ 1 2 ˆ ˆ j C rx n C w 2˛ E ˆ ˆ ˆ ı ı ı ˆ ˆ ˆ ˆ 2 2ˇ 2˛ ˆ ˆ D E C ^ B u ˆ 2 @t j ˆ ˆ ı ı ı ˆ ˆ Z ˆ ˆ 1 C ˆ ˆ div g …gC g C …g M dv ˆ ˆ ˆ ı R3 ˆ Z ˆ ˆ ˆ ˆ C C C ˆ L hC ˆ …h h C …h ; h …h h C …h vM dv ˆ ˆ R3 ˆ Z ˆ < C

Cı Q g ; g Q g ; gC vM dv; ˆ R3 ˆ ˆ ˆ 1 ˆ ˆ ˆ w C 2 div j ˆ ˆ

ı ˆ ˆ ˆ ˆ 2˛ 3 2 ˆ ˆ u D E @t w ˆ ˆ ˆ ı 2ı2 ˆ Z ˆ ˆ 1 C ˆ ˆ ˆ div g …gC g C …g M dv ˆ ˆ ı ˆ R3 ˆ Z ˆ ˆ C 2 ˆ ˆ C C C jvj ˆ L …h h C …h ; h …h h C …h M dv h ˆ ˆ 2 ˆ R3 ˆ ˆ Z ˆ ˆ C jvj2 ˆ ˆ Q g ; g Q g ; gC M dv: Cı : 2 R3 (2.77)

64


In fact, the system (2.77) coupled with Maxwell’s equations (2.51) still contains a singular perturbation, which will be treated much like the singular perturbation of the one-species case in Sections 2.3.2 and 2.3.3. Thus, by virtue of (2.45), it is readily seen that the system (2.77) may be further simplified to 8 div j D O.ı/; ˆ ˆ ˆ ˆ ˆ 2˛ ˆ ˆ C w D E C O.ı/; r n < x ı ˇ ˛ ˆ ˆ Pj D P E C u ^ B C o.1/; ˆ ˆ ˆ ı ı ˆ ˆ : w D o.1/:

(2.78)

We discuss now the limit ! 0 of the coupled system (2.51)–(2.78). (1) When 1 (so that ˛ D O.ı/), letting ! 0 in (2.51)–(2.78), we obtain 8 ˇ ˆ ˆ j; div E D 0; Œ @t E rot B D ˆ ˆ ı ˆ ˆ ˆ < Œ @ B C rot E D 0; div B D 0; t h i ˛ ˇ ˆ ˆ ˆ j D rx pN C EC u ^ B ; div j D 0; ˆ ˆ ı ı ˆ ˆ : n D 0; w D 0; where pN is an electrodynamic pressure. (2) When D o.1/ and ˛ D O.ı/, letting ! 0 in (2.51)–(2.78), we obtain 8 E D 0; ˆ ˆ ˆ ˆ ˆ ˇ ˆ ˆ j; div B D 0; < rot B D ı ˇ ˆ ˆ ˆ j D rx pN C u ^ B ; div j D 0; ˆ ˆ ı ˆ ˆ : n D 0; w D 0; where pN is an electrodynamic pressure. ˛ is unbounded, we need to further use Faraday’s equa(3) When D o.1/ and ı tion from (2.51), as in Section 2.3.3, to write that

˛ ˇ @t A C PE D 0; ı ı


65

where B D rot A and div A D 0. Thus, letting ! 0 in (2.51)–(2.78), we obtain 8 h˛ i ˆ n; rot E D 0; div E D ˆ ˆ ˆ ˆ ˆ ˆ ˇ ˆ ˆ ˆ div B D 0; ˆ rot B D ı j; ˆ < rot A D B; div A D 0; ˆ ˆ ˆ ˇ ˇ ˆ ˆ j D rx pN @t A C u ^ B ; div j D 0; ˆ ˆ ˆ ı ı ˆ i h ˆ ˆ ˛ ˆ : rx n D 2 E; w D 0; where 2 A ! A and pN is an electrodynamic pressure. Note that x n D 2 ˛ n, so that necessarily n D 0 and E D 0. The above system can be rewritten more explicitly by defining the adjusted electric field EQ D @t A. It then holds that 8 ˇ ˆ ˆ j; div B D 0; rot B D ˆ ˆ ı ˆ ˆ ˆ < @ B C rot EQ D 0; div EQ D 0; t ˆ ˇ Q ˇ ˆ ˆ j D rx pN C u ^ B ; div j D 0; EC ˆ ˆ ı ı ˆ ˆ : n D 0; w D 0:

2.4.6 Summary At last, we see that the asymptotics of the two-species Vlasov–Maxwell–Boltzmann system (2.36) can be depicted in terms of the limits of the following parameters: the strength of the electric induction ˛, the strength of the magnetic induction ˇ D

˛ ,

the ratio of the bulk velocity to the speed of light , the strength of the interspecies collisional interactions ı. The case of very weak interspecies collisions has already been discussed in Section 2.4.2 and is analogous to the one-species case. Regarding the weak and strong interspecies collisions, Figures 2.2 and 2.3 summarize the different asymptotic regimes, on a logarithmic scale, of the two-species Vlasov–Maxwell–Boltzmann system (2.36). Thus, up to multiplicative constants, in the case of strong interactions ı D 1, we reach the following asymptotic systems of equations:

66


(1) If ˛ D o./, we obtain the two-fluid incompressible resistive Navier–Stokes– Fourier system 8 div u D 0; @t u C u rx u x u D rx p; ˆ ˆ < @t C u rx x D 0; C D 0; (2.79) ˆ ˆ : @t n C u rx n x n D 0; j nu D rx n; w D n: 2 2 This system satisfies the following formal energy conservation laws: 1 d kuk2L2 C krx uk2L2 D 0; x x 2 dt 1 d k k2L2 C krx k2L2 D 0; x x 2 dt 1 d 2 2 knkL2 C krx nkL2 D 0: x x 2 dt 2 (2) If ˛ D and D 1, we obtain the two-fluid incompressible Navier–Stokes– Fourier–Maxwell system with Ohm’s law 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C .nE C j ^ B/ ; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; div E D n; @t E rot B D j; ˆ ˆ ˆ @t B C rot E D 0; div B D 0; ˆ ˆ ˆ ˆ 1 ˆ ˆ : w D n: j nu D rx n C E C u ^ B ; 2 (2.80) This system satisfies the following formal energy conservation laws (see Proposition 3.3 for an explicit computation of the energy): 1 d 1 2 2 2 2 2 kukL2 C knkL2 C kEkL2 C kBkL2 C krx uk2L2 x x x x x 4 dt 2 1 C kj nuk2L2 D 0; x 2 1 d k k2L2 C krx k2L2 D 0: x x 2 dt (3) If ˛ D and D o.1/, we obtain the two-fluid incompressible Navier– Stokes–Fourier–Poisson system with Ohm’s law 8 1 ˆ ˆ @t u C u rx u x u D rx p C nrx ; div u D 0; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; @t C u rx x D 0; < (2.81) @t n C u rx n x n C n D 0;

x D n; ˆ ˆ ˆ 2 ˆ ˆ ˆ 1 ˆ ˆ : j nu D rx n ; w D n: 2

2.4 Formal analysis of the two-species asymptotics two-fluid incompressible Navier-Stokes-Fourier-Maxwell with Ohm´s law ϵ

1

ϵ

ϵ2 two-fluid incompressible Navier-Stokes-Fourier-Poisson with Ohm´s law

67 γ

ϵ2 two-fluid incompressible resistive Navier-Stokes-Fourier

two-fluid incompressible resistive Navier-Stokes-Fourier ϵ3

α

Figure 2.2. Asymptotic regimes of the two-species Vlasov–Maxwell–Boltzmann system (2.36) for strong interspecies interactions on a logarithmic scale.

This system satisfies the following formal energy conservation laws: 1 1d 2 2 2 2 kukL2 C knkL2 C krx kL2 C krx uk2L2 x x x x 4 dt 2 1 C kj nuk2L2 D 0; x 2 1d knk2L2 C krx nk2L2 C knk2L2 D 0; x x x 2 dt 2 1d k k2L2 C krx k2L2 D 0: x x 2 dt Finally, up to multiplicative constants, in the case of weak interactions ı D o.1/, we reach the following asymptotic systems of equations: (1) If ˇ D o.ı/, we obtain the incompressible Navier–Stokes–Fourier system 8 ˆ < @t u C u rx u x u D rx p; div u D 0; C D 0; @t C u rx x D 0; (2.82) ˆ : n D 0; j D 0; w D 0: This system satisfies the following formal energy conservation laws: 1 d kuk2L2 C krx uk2L2 D 0; x x 2 dt 1 d k k2L2 C krx k2L2 D 0: x x 2 dt

68


(2) If ˛ D ı and D 1, we obtain the two-fluid incompressible Navier–Stokes– Fourier–Maxwell system with solenoidal Ohm’s law 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C j ^ B; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; @t E rot B D j; div E D 0; (2.83) ˆ ˆ div B D 0; @t B C rot E D 0; ˆ ˆ ˆ ˆ ˆ j D .rx pN C E C u ^ B/ ; div j D 0; ˆ ˆ : n D 0; w D 0: This system satisfies the following formal energy conservation laws (see Proposition 3.3 for an explicit computation of the energy):

1 1 d 2 kuk2L2 C kEk2L2 C kBk2L2 C krx uk2L2 C kj k2L2 D 0; x x x x x 4 dt 2 1 d k k2L2 C krx k2L2 D 0: x x 2 dt (3) If ˇ D ı and D o.1/, we obtain the two-fluid incompressible quasi-static Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C j ^ B; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; rot B D j; div E D 0; (2.84) ˆ ˆ div B D 0; @t B C rot E D 0; ˆ ˆ ˆ ˆ ˆ j D .r p N C E C u ^ B/ ; div j D 0; ˆ x ˆ : n D 0; w D 0: This system satisfies the following formal energy conservation laws:

1 1d 2 kuk2L2 C kBk2L2 C krx uk2L2 C kj k2L2 D 0; x x x x 4 dt 2 1 d k k2L2 C krx k2L2 D 0: x x 2 dt Here, the electric field is defined indirectly as a mere distribution, through Faraday’s equation, by E D @t A;


69

where B D rot A and div A D 0. Note that the above system can be rewritten as 8 1 ˆ ˆ @t u C u rx u x u D rx p C rot B ^ B; div u D 0; ˆ ˆ < 2 C D 0; @t C u rx x D 0; ˆ ˆ ˆ 1 ˆ : @t B C u rx B x B D B rx u; div B D 0; which is nothing but the well-known magnetohydrodynamic system. The rigorous derivation of this system starting from other macroscopic systems such as (2.53) and (2.83) has been investigated in [6]. δ

1 magnetohydrodynamics

δ2

γ incompressible Navier-Stokes-Fourier

ϵ

two-fluid incompressible Navier-Stokes-Fourier-Maxwell with solenoidal Ohm´s law

magnetohydrodynamics

δϵ

incompressible Navier-Stokes-Fourier incompressible Navier-Stokes-Fourier δ2ϵ α

Figure 2.3. Asymptotic regimes of the two-species Vlasov–Maxwell–Boltzmann system (2.36) for weak interspecies interactions on a logarithmic scale.

2.4.7 The two-species Vlasov–Poisson–Boltzmann system The two-species Vlasov–Poisson–Boltzmann system describes the evolution of a gas of two species of charged particles (cations and anions) subject to a self-induced electrostatic force. This system is obtained formally from the two-species Vlasov– Maxwell–Boltzmann system by letting the speed of light tend to infinity while all other parameters remain fixed. Accordingly, setting D 0 in (2.36) yields the scaled

70


Vlasov–Poisson–Boltzmann system: 8 1 ı2 ˆ ˙ ˙ ˙ ˙ ˙ ˆ ˆ f C v r f ˙ ˛r r f D ; f / C Q.f Q.f˙ ; f /; @ t x x v ˆ ˆ < (2.85) f˙ D M 1 C g˙ ; ˆ Z ˆ ˆ C ˛ ˆ ˆ g g M dv:

x D : R3 Here, the plasma is subject to a self-induced electrostatic field E determined by Z C ˛ rot E D 0; div E D g g M dv; R3 hence

E D rx :

The above system is supplemented with some initial data satisfying Z 1 in 1 1 Cin H f H f jE in j2 dx < 1: C C 2 2 2 R3 In particular, solutions of (2.85) satisfy the corresponding scaled entropy inequality, where t > 0, Z 1 C 1 1 H f H f jE j2 dx C C 2 2 2 R3 Z Z C

1 t D f C D f C ı2 D fC ; f .s/ dxds C 4 0 R3 Z 1 1 1 jE in j2 dx: 2 H fCin C 2 H fin C 2 R3 Thus, the formal asymptotic analysis of (2.85) is contained in our analysis of the two-species Vlasov–Maxwell–Boltzmann system (2.36). Specifically, setting D ˇ D 0 in the limiting systems first obtained in Section 2.4.2, for very weak interspecies collisions, we see that the two-species Vlasov–Poisson–Boltzmann system (2.85) converges, when ˛ D o./ and ı D O./, towards the two-fluid incompressible Navier–Stokes–Fourier system in a Boussinesq regime, with E D 0, 8 2 ˆ 1 C ı ˆ ˙ ˙ ˙ ˙ ˆ @t u C u rx u x u ˙ u u D rx p˙ ; div u˙ D 0; < 2 ˆ ı 1 C ˆ ˆ ˙ C ˙ D 0: D 0; : @t ˙ C u˙ rx ˙ x ˙ ˙


71

On the other hand, when ˛ ¤ 0 and ı D O./, we have convergence towards the two-fluid incompressible Navier–Stokes–Fourier–Poisson system 8 2 ˆ 1 C ı ˆ ˙ ˙ ˙ ˙ ˆ u u D rx p˙ C ˙ rx ˙ ; @t u C u rx u x u ˙ ˆ ˆ ˆ ˆ ˆ ˆ ˙ ˆ D 0; div u ˆ ˆ ˆ ˆ < 5 3 ˙ 3 ˙ ˙ ˙ ˙ C u rx x ˙ @t 2 2 2 ˆ ˆ 2 ˆ ˆ 5 1 ı ˆ ˆ ˆ ˙ C D 0; ˆ ˆ 2

ˆ ˆ h i2 ˆ ˆ ˆ : x ˙ C ˙ D ˙ ˛ C ;

where the electrostatic field is determined by ˛ E D ˙rx .˙ C ˙ /. Regarding weak interspecies interactions, setting D ˇ D 0 in the corresponding limiting systems obtained in Sections 2.4.4 and 2.4.5, we see that the two-species Vlasov–Poisson–Boltzmann system (2.85) always converges, when ı D o.1/ and ı is unbounded, towards the incompressible Navier–Stokes–Fourier system in a Boussinesq regime, with E D 0 8 ˆ < @t u C u rx u x u D rx p; div u D 0; @t C u rx x D 0; C D 0; ˆ : n D 0; j D 0; w D 0: Finally, in the case of strong interspecies interactions, setting D ˇ D 0 in the corresponding limiting systems obtained in Sections 2.4.4 and 2.4.5, we see that the two-species Vlasov–Poisson–Boltzmann system (2.85) converges, when ˛ D o./ and ı D 1, towards the two-fluid incompressible resistive Navier–Stokes–Fourier system in a Boussinesq regime, with E D 0 8 div u D 0; @t u C u rx u x u D rx p; ˆ ˆ < C D 0; @t C u rx x D 0; ˆ ˆ : @t n C u rx n x n D 0; j nu D rx n; w D n: 2 2 ˛

On the other hand, when ¤ 0 and ı D 1, we find the convergence towards the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm’s law 8 1 h˛ i ˆ ˆ nrx ; div u D 0; u C u r u u D r p C @ t x x x ˆ ˆ 2 ˆ ˆ ˆ ˆ @t C u rx x D 0; C D 0; < h ˛ i2 h˛ i n; @ n C u r n n C n D 0;

D

ˆ t x x x ˆ ˆ 2 ˆ ˆ h˛ i ˆ 1 ˆ ˆ : j nu D rx n ; w D n; 2 where the electrostatic field is determined by E D rx .

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In fact, the two-species Vlasov–Poisson–Boltzmann system is inherently simpler than the two-species Vlasov–Maxwell–Boltzmann system, because it couples the Vlasov–Boltzmann equations with a simple elliptic equation, namely Poisson’s equation, while the two-species Vlasov–Maxwell–Boltzmann system couples the Vlasov– Boltzmann equations with an hyperbolic system, namely the Maxwell system of equations. Thus, the rigorous mathematical analysis on the two-species Vlasov– Maxwell–Boltzmann system, presented in the remainder of this work, will also apply to the two-species Vlasov–Poisson–Boltzmann system and, therefore, analogous results will hold.

Chapter 3

Weak stability of the limiting macroscopic systems In the previous chapter, we have formally derived numerous viscous incompressible systems for plasmas starting from Vlasov–Maxwell–Boltzmann systems and we intend to provide, in the remainder of our work, justifications of these derivations. Nevertheless, prior to any rigorous proof of hydrodynamic limit, it is crucial to understand the well-posedness of the asymptotic macroscopic models and to study their stability properties. Describing the Cauchy problem of each single macroscopic system from Chapter 2 would be unreasonable. Rather, we are now going to focus on the following three systems found therein: the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (2.33); the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law (2.80); the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law (2.83), and establish the existence of weak or dissipative solutions to their respective initial value problems. In fact, these three systems are among the most singular ones found in Chapter 2. Thus, we hope the reader will find it clear that the existence of appropriate weak or dissipative solutions to the remaining macroscopic systems from Chapter 2 will then follow from straightforward adjustments of the existence theories presented here. In the remaining Parts II, III and IV of our work, we will also focus on the three aforementioned systems and give complete justifications of their derivation from hydrodynamic limits of Vlasov–Maxwell–Boltzmann systems.

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3 Weak stability of the limiting macroscopic systems

3.1 The incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system We are first concerned here with the incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system (2.33), which we rewrite, for mere convenience, as 8 @t u C u rx u x u D rx p C E C rx C u ^ B; ˆ ˆ ˆ ˆ ˆ div u D 0; ˆ ˆ ˆ < 3 5 3 @t

x . C / D ; C u rx x D 0; ˆ 2 2 2 ˆ ˆ ˆ ˆ rot B D u; div E D ; ˆ ˆ ˆ : div B D 0: @t B C rot E D 0; (3.1) Although it looks more complicated because it involves more terms, the system (3.1) has the same structure as the incompressible Navier–Stokes equations: it is indeed a system of parabolic equations, in which the nonlinear advection terms are welldefined by the energy estimate. The following formal proposition shows how to compute the energy. Proposition 3.1. Let .; u; ; B/ be a smooth solution to the incompressible quasistatic Navier–Stokes–Fourier–Maxwell–Poisson system (3.1). Then, the following global energy inequality holds: 3 1 2 2 2 2 2 k.t/kL2 C ku.t/kL2 C k.t/kL2 C krx . C / .t/kL2 C kB.t/kL2 x x x x x 2 2 Z t 5 krx u.s/k2L2 C krx .s/k2L2 ds C x x 2 0 1 in 2 2 C uin 2 2 C 3 in 2 2 C rx in C in 2 2 C B in 2 2 ; Lx Lx Lx Lx Lx 2 2 (3.2) where rx . C / D P ? E ¤ E. Proof. Multiplying the equation expressing the conservation of momentum in (3.1) by u and integrating with respect to space variables, we get Z 1 d 2 2 .u E C u rx / dx kukL2 C krx ukL2 D x x 3 2 dt ZR D .rot B E C u rx / dx R3 Z 1 d 2 u rx dx: D kBkL2 C x 2 dt R3

3.1 The incompressible quasi-static Navier–Stokes–Fourier–. . .

75

Then, multiplying the equation expressing the conservation of energy by , we similarly get Z 3 d 5 2 2 . @t C u rx / dx k kL2 C krx kL2 D x x 4 dt 2 R3 Z 1 d 2 Œ. C / @t u rx dx D kkL2 C x 2 dt R3 1 d 1 d D kk2L2 krx . C / k2L2 x x 2 dt 2 dt Z R3

u rx dx:

Summing the above identities, we obtain the expected global conservation of energy. Based on the functional spaces naturally suggested by the energy inequality (3.2) from Proposition 3.1, we provide now a suitable notion of weak solution for the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (3.1) and establish the existence of such solutions next. This definition is modeled after the work of Leray [50] establishing the existence of weak solutions (or Leray solutions) of the incompressible Navier–Stokes equations. We refer to [48, 49] for a modern treatment of the incompressible Navier–Stokes equations. Henceforth, we will utilize the prefixes w- or w - to express that a given space is endowed with its weak or weak- topology, respectively. Also, recall that P W L2 .dx/ ! L2 .dx/ is the Leray projector onto solenoidal vector fields, that is, P D 1 rot rot. Definition. We say that

.; u; ; B/ 2 L1 Œ0; 1/I L2 R3 \ C Œ0; 1/I w-L2 R3 ;

such that

.u; / 2 L2 Œ0; 1/I HP 1 R3 ;

is a weak solution (or Leray solution) of the incompressible quasi-static Navier– Stokes–Fourier–Maxwell–Poisson system (3.1), if it solves the system 8 @t .u C A/ C P .u rx u/ x u D P .rx C u ^ B/ ; ˆ ˆ ˆ ˆ ˆ div u D 0; < 5 3 3 ˆ

x . C / D ; @ C u rx x D 0; ˆ ˆ t 2 2 2 ˆ ˆ : rot B D u; div B D 0; in the sense of distributions, where A 2 L1 .Œ0; 1/I HP 1.R3 // is the unique solution to rot A D B; div A D 0; and if the energy inequality (3.2) is verified.

76


Using the a priori estimates provided by the energy inequality (3.2) and reproducing the arguments of Leray [50], we can easily establish the global existence of weak solutions. Indeed, combining first the bound on rx . C / with the additional spatial regularity on u and , coming from the dissipation terms in the energy inequality (3.2), we infer that all three terms , u and enjoy some spatial regularity. More precisely, they are all uniformly bounded in L2loc .dtI H 1 .dx//. Furthermore, recalling that PE D @t A, some temporal regularity on u C A and 32 is clearly inherited from the evolution equations, which allows us to establish, invoking a classical compactness result by Aubin and Lions [9, 52] (see [73] for a sharp compactness criterion; see also [49, Section 12.1]), that u C A and 32 are strongly relatively compact in all variables in L2loc .dtdx/. Finally, noticing that one can write x 2 x 3 .u C A/ ; ; uD D 3 5 x 2 1 x 2 2 x 3 rot D .u C A/ ; ; B D 3 5 x 2 1 x using the Poisson equations x A D u;

x . C / D ;

we easily find that all four observables , u, and B belong to a compact subset of L2loc .dtdx/. The above compactness properties allow us to prove the weak stability of the nonlinear terms in (3.1) and, therefore, to take weak limits in any suitable approximation scheme in order to establish the existence of weak solutions. This is precisely the content of the coming theorem. Theorem 3.2. For any initial data in ; uin ; in ; B in 2 L2 .R3 / such that div uin D 0; div B in D 0; rot B in D uin ;

x in C in D in ; there exists a weak solution to the incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system (3.1). As usual for such weak solutions, uniqueness is not known to hold. To prove that the system (3.1) is well-posed in the sense of Hadamard, i.e., that solutions exist, are unique, and depend continuously on the initial data, we would have to deal with a stronger notion of solution. Note however that, by modulating the energy inequality, we can establish some weak-strong uniqueness principle, meaning that if a somewhat regular solution to (3.1) is known to exist, then any weak solution with matching initial data coincides with the smooth one as long as it exists. We refer to the next Section 3.2 for details on how to modulate the energy and, thus, establish such weakstrong uniqueness principles. Analogous existence results, based on the same preceding compactness arguments, hold for the similar systems (2.32), (2.34), (2.54), (2.55) and (2.56), as well. For the sake of conciseness, we refrain from stating these existence results here.

3.2 The two-fluid incompressible Navier–Stokes–Fourier–Maxwell. . .

77

As for the remaining systems (2.30), (2.31), (2.52), (2.53), (2.79), (2.81), (2.82) and (2.84) – that is to say, all remaining macroscopic systems from Sections 2.3.4, 2.4.2 and 2.4.6, except (2.80) and (2.83) – the existence of their corresponding weak solutions is established through similar, and even simpler, standard arguments which go back to the original work of Leray [50]. We refer to [48, 49] for a modern reference on classical results on Leray solutions for the incompressible Navier–Stokes equations providing a complete toolbox for establishing the existence of such weak solutions. Finally, concerning systems (2.80) and (2.83), these are highly singular and the preceding compactness arguments are insufficient to infer their weak stability in corresponding energy spaces. In fact, weak solutions are not know to exist in full generality. These issues are discussed in the next section.

3.2 The two-fluid incompressible Navier–Stokes–Fourier– Maxwell system with (solenoidal) Ohm’s law We focus now on the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law (2.80), 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C .nE C j ^ B/ ; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; @t E rot B D j; div E D n; (3.3) ˆ ˆ ˆ B C rot E D 0; div B D 0; @ t ˆ ˆ ˆ ˆ 1 ˆ ˆ : w D n; j nu D rx n C E C u ^ B ; 2 and on the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law (2.83), 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C j ^ B; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; div E D 0; @t E rot B D j; (3.4) ˆ ˆ @ B C rot E D 0; div B D 0; ˆ t ˆ ˆ ˆ ˆ j D .rx pN C E C u ^ B/ ; div j D 0; ˆ ˆ : n D 0; w D 0: The models (3.3) and (3.4) are not stable under weak convergence in the energy space and, thus, share more similarities with the three-dimensional incompressible Euler equations, as we are about to discuss.

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To this end, note first that the advection-diffusion equation on is not really coupled with the other equations on .u; n; j; E; B/ in (3.3) and (3.4), and that it is linear provided the velocity field u is given. It is therefore sufficient to focus on the reduced systems of equations 8 1 ˆ ˆ @t u C u rx u x u D rx p C .nE C j ^ B/ ; div u D 0; ˆ ˆ 2 ˆ ˆ ˆ < div E D n; @t E rot B D j; (3.5) div B D 0; @t B C rot E D 0; ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ n C E C u ^ B ; j nu D r : x 2 and 8 1 ˆ div u D 0; ˆ @t u C u rx u x u D rx p C j ^ B; ˆ ˆ 2 < div E D 0; @t E rot B D j; ˆ ˆ div B D 0; @t B C rot E D 0; ˆ ˆ : j D .rx pN C E C u ^ B/ ; div j D 0:

(3.6)

Remark. The system (3.6) can be viewed as an asymptotic regime of the system (3.5). Indeed, at least formally, it is obtained, as ı ! 0, from the system 8 1 ˆ div u D 0; @t u C u rx u x u D rx p C .ınE C j ^ B/ ; ˆ ˆ ˆ 2 ˆ ˆ < @t E rot B D j; div E D ın; @ B C rot E D 0; div B D 0; ˆ t ˆ ˆ ˆ ˆ 1 ˆ : j ınu D rx n C E C u ^ B ; 2ı which is consistent with the formal derivations from Section 2.4. A natural framework to study these equations (coming from physics) should be the energy space, i.e., the functional space defined by the (formal) energy conservation. We indeed expect solutions in this space to be global. The following formal proposition shows how to compute the energy of the twofluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), or with solenoidal Ohm’s law (3.6). Proposition 3.3. Let .u; E; B/ be a smooth solution to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), or with solenoidal Ohm’s law (3.6). Then the following global conservation of energy holds: Z t E .t/ C D .s/ ds D E .0/; for all t > 0; 0


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where the energy E and the energy dissipation D are given, for the system (3.5), by 1 1 1 1 ku.t/k2L2 C kn.t/k2L2 C kE.t/k2L2 C kB.t/k2L2 ; x x x x 2 8 4 4 1 D .t/ D krx u.t/k2L2 C k.j nu/ .t/k2L2 ; x x 2

E .t/ D

or, for the system (3.6), by 1 1 1 ku.t/k2L2 C kE.t/k2L2 C kB.t/k2L2 ; x x x 2 4 4 1 D .t/ D krx u.t/k2L2 C kj.t/k2L2 : x x 2

E .t/ D

Proof. We consider the system (3.5) first. Multiplying the equation expressing the conservation of momentum in (3.5) by u and integrating with respect to the space variables, we get Z 1d 1 kuk2L2 C krx uk2L2 D .nE C j ^ B/ u dx; x x 2 dt 2 R3 while multiplying Ohm’s law in (3.5) by j nu and integrating in space yields the identity Z 1 1 2 n div j C E j .nE C j ^ B/ u dx; kj nukL2 D x 2 R3 where we have employed the incompressibility of the velocity field. Hence, we obtain, exploiting further the continuity equation @t n C div j D 0 (deduced by taking the divergence of Ampère’s equation and from Gauss’ law), that Z 1 1 1 d 1 2 2 2 kukL2 C krx ukL2 C n div j C E j dx kj nukL2 D x x x 2 dt 2 4 2 R3 Z 1 1 d E j dx: D knk2L2 C x 8 dt R3 2 As for the system (3.6), similar, and actually simpler computations yield that Z 1 1 1 d kuk2L2 C krx uk2L2 C E j dx: kj k2L2 D x x x 2 dt 2 R3 2 Next, for both systems (3.5) and (3.6), the conservation of the electromagnetic energy is given by the Maxwell equations Z

1 d E j dx: kEk2L2 C kBk2L2 D x x 2 dt R3 Summing the above formal identities leads to the claimed global conservation of energy.

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The uniform bounds resulting from the energy conservations in Proposition 3.3 imply that all the terms in systems (3.5) and (3.6) make sense, especially the nonlinear terms in the motion equations and in Ohm’s laws. Notice, however, that it is at first not clear that the Lorentz force nE C j ^ B in (3.5) is a well-defined distribution, based on the natural a priori estimates provided by the energy and energy dissipation, because j does not necessarily lie in L1t L2x . Nevertheless, it is possible to give it a rigorous sense by exploiting simple identities. A first approach consists in identifying the force term nE C j ^ B with the conservation law for the electromagnetic energy flux E ^ B (also called the Poynting vector, see [45, Section 6.7]) 1 @t .E ^ B/ C rx E 2 C B 2 rx .E ˝ E C B ˝ B/ D nE j ^ B; (3.7) 2 derived directly from Maxwell’s equations in (3.5) (see the derivation of (1.11) and (1.12)), so that the force makes sense in some Sobolev space with negative regularity index. In fact, it will be much more appropriate to estimate the Lorentz force directly using Ohm’s law from (3.5) as follows nE C j ^ B D .j nu/ ^ B C n .E C u ^ B/ 1 1 D .j nu/ ^ B C n .j nu/ C rx n2 ; 4 so that the force is now understood as the sum of a locally integrable function and a pressure gradient. All other terms from (3.5) and (3.6) are obviously well-defined. Unfortunately, the uniform energy bounds do not guarantee the weak stability of the nonlinear terms nE and j ^ B composing the Lorentz force. This is a major obstacle to establishing the global existence of weak solutions in the spirit of Leray [50], which are therefore not known to exist in general. There are two evident strategies, which unfortunately turn out to be unsuccessful, that one would want to apply here in order to circumvent the lack of weak stability of the Lorentz force in systems (3.5) and (3.6). The first strategy consists in propagating strong compactness or regularity in Maxwell’s equations, which are indeed the archetype of hyperbolic equations, meaning that singularities are propagated. In general, these singularities, or oscillations, may be created either by boundary data, by initial data, or by the source terms, and they remain localized on the corresponding light cones. Here, we are not considering boundaries and the initial data can always be well-prepared. However, it remains unclear how to prevent the emergence of oscillations from the source term j in Maxwell’s equations, which is determined by the nonlinear Ohm laws in (3.5) and (3.6). Therefore, we do not expect to gain regularity (or even compactness) on the electromagnetic field .E; B/. So, this strategy fails in general. It is to be noted, though, that this approach has been successfully applied by Masmoudi [60] to a slightly different system coupling the incompressible Navier–Stokes equations with Maxwell’s equations in the two-dimensional case. Since the equations


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studied in [60] are very similar to (3.5) and (3.6), we present Masmoudi’s result below in Section 3.2.1 in order to emphasize the mathematical difficulties inherent to the coupling with Maxwell’s equations through Ohm’s law and its similarities with the two-dimensional Euler equations. Also, we believe that similar results on systems (3.5) and (3.6) can be achieved. The second strategy consists in utilizing the linear structure of the Maxwell equations with the specific quadratic structure of the Lorentz force to apply the theory of compensated compactness of Murat and Tartar [65, 66, 75] (see also [74] for an introduction to the subject) and, thus, filter any undesired nonlinear resonances. This approach plainly fails and it seems that it can only potentially succeed by exploiting the full nonlinear structure of the whole systems (3.5) and (3.6). But we are not aware of such successful nonlinear treatment of resonances. We refer to [6] for some more details about the failure of the method of compensated compactness in the electromagnetic setting. Following the concise Section 3.2.1 below, where we present the main result from [60] on the well-posedness of an incompressible Navier–Stokes–Maxwell system in two dimensions, we will discuss very briefly in Section 3.2.2 the well-posedness of the same system in three dimensions and for small initial data. Finally, in Section 3.2.3, we will introduce the dissipative solutions of the systems (3.5) and (3.6) and justify their global existence in any dimension, which will be particularly relevant to our work.

3.2.1 Large global solutions in two dimensions In [60], Masmoudi studied the following incompressible Navier–Stokes–Maxwell system: 8 ˆ < @t u C u rx u x u D rx p C j ^ B; div u D 0; j D .E C u ^ B/ ; @t E rot B D j; (3.8) ˆ : @t B C rot E D 0; div B D 0; which is somewhat related to the systems (3.5) and (3.6), and satisfies the formal energy conservation

1 1 d (3.9) kuk2L2 C kEk2L2 C kBk2L2 C krx uk2L2 C kj k2L2 D 0: x x x x x 2 dt Notice that, in this system, there is no constraint on div E or div j . He restricted his analysis to the two-dimensional case, which is obtained by assuming that 0 1 1 0 1 0 u1 .x1 ; x2 / E1 .x1 ; x2 / 0 A: 0 u D @u2 .x1 ; x2 /A ; E D @E2 .x1 ; x2 /A and B D @ 0 0 B3 .x1 ; x2 / In order to understand the propagation of singularities in Maxwell’s system (in two or three dimensions), it is often convenient to express it using vector and scalar

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potentials in an equivalent form (see [45, Sections 6.2 and 6.3]). To this end, since the magnetic field B is solenoidal, we may always write B D rot A, for some vector potential A. Moreover, taking into account Faraday’s equation, we see that necessarily E D rx ' @t A for some scalar potential '. As a matter of fact, the potentials A and ' are not uniquely determined. Indeed, the electromagnetic field is invariant under the so-called gauge transformation .A; '/ 7! .A C rx ; ' @t /. This gauge invariance allows us to impose a further condition on the potentials. Typically, one may impose the so-called Coulomb gauge div A D 0, which is simple and natural for stationary settings. Another classical example of gauge fixing includes the Lorenz (not to be confused with Lorentz) gauge div A D @t ', which usually yields an evolution for the potentials governed by decoupled wave equations. Here, for the Maxwell system in (3.8), we choose the slight variant of the Lorenz gauge div A D @t ' '; (3.10) which yields the decoupled damped wave equation @2t A C @t A x A D u ^ .rot A/ :

(3.11)

Note that it is always possible to find A and ' satisfying (3.10). Indeed, if (3.10) is not satisfied, one may always apply a gauge transformation with solving the damped wave equation @2t C @t x D div A C @t ' C ' and produce new potentials for which (3.10) holds. Now, if the velocity field u is bounded in L1 .Œ0; T ; dtI L1 .dx//, for some T > 0, it is possible to show, through standard energy estimates, that the damped wave equation (3.11), which is linear in A, propagates the strong compactness of @t A and rx A in L1 .Œ0; T ; dtI L2 .dx//. More precisely, considering a sequence of vector fields fAn gn2N solving (3.11), such that f@t An .t D 0/gn2N and frx An .t D 0/gn2N are Cauchy sequences in L2 .dx/ (or, by possibly extracting subsequences, that they lie in a compact subset), we find, performing a standard energy estimate on (3.11), that Z Z

1 d 2 2 j@t .Am An /j2 dx j@t .Am An /j C jrx .Am An /j dx C 2 dt R3 R3 Z D .u ^ rot .Am An // @t .Am An / dx R3 Z

ku.t/kL1 .dx/ j@t .Am An /j2 C jrx .Am An /j2 dx: 2 R3 An application of Grönwall’s lemma then yields that Z

j@t .Am An /j2 C jrx .Am An /j2 dx R3 Z

2 kuk 1 1 Lt Lx e j@t .Am An / .t D 0/j2 C jrx .Am An / .t D 0/j dx; R3


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whence f@t An gn2N and frx An gn2N are convergent in L1 .Œ0; T ; dtI L2 .dx//. This would obviously imply the propagation of strong compactness for the magnetic field B. Unfortunately, in two dimensions of space, the H 1 estimate on the velocity field u provided by the conservation of energy barely fails to yield, by Sobolev embedding, an L1 bound on u. Masmoudi’s idea was then to compensate this lack of critical embedding by placing the initial electromagnetic field in a better H s space, with 0 < s < 1, and to propagate this initial regularity with Maxwell’s equations at the same time that the parabolic regularity of the Stokes flow is employed to estimate the velocity field in a higher regularity space. This approach eventually allows to bound 2 P1 u in L1t L1 x in terms of its Lt Hx norm with some logarithmic loss. As a byproduct of these estimates, it is also possible to establish the exponential growth of the H s norms. In Masmoudi’s own words: “One can compare this growth estimate with the double exponential growth estimate of the H s norms in the twodimensional incompressible Euler system” [60]. Finally, it is interesting to note that Masmoudi’s proof uses neither the divergence free condition of the magnetic field nor the decay property of the linear part coming from Maxwell’s equations. The following theorem contains the main well-posedness result from [60]. Note that it gives the existence and uniqueness for initial data in a very large dense subspace S of L2 , namely in 0<s 0, u 2 C Œ0; T I L2 \ L2 Œ0; T I HP 1 and E; B 2 C Œ0; T I H s : Moreover,

j 2 L2 Œ0; T I L2 \ L1 Œ0; T I H s

and

0 u 2 L1 Œ0; T I H s ;

for each 1 < s 0 < min .2s C 1; 2/. In addition, the energy identity (3.9) holds and we have the following exponential growth estimate for all t > 0: in s C B in s eC in .1Ct / ; kuk 0 C kE.t/k s C kB.t/k s 1 C E L1 Œ0;t IH s

H

H

H

H

2 2 2 where C in D C 1 C uin L2 C E in L2 C B in L2 for some constant C .

3.2.2 Small global solutions in three dimensions As we have seen, there are serious obstacles to the construction of global solutions of the system (3.8) for large initial data in the energy space. Nevertheless, it is in

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general possible to establish the well-posedness of a system, globally in time, by showing its strong stability for small initial data in some space satisfying the same scaling invariance as the given system of equations. This is precisely what Ibrahim and Keraani managed to achieve in [43] for the three-dimensional incompressible Navier–Stokes–Maxwell system (3.8) using the strategy of Fujita and Kato [35], which is based on refined a priori estimates obtained by paradifferential calculus and some fixed-point argument. Note that the results from [43] do not imply the local existence of strong solutions for large data, which has been established in a separate work by Ibrahim and Yoneda in [44]. These results have then been unified and extended to a more natural setting by Germain, Ibrahim and Masmoudi in [36]. We believe that the methods employed in [36, 43, 44] can potentially lead to similar results for the analogous incompressible Navier–Stokes–Maxwell systems (3.5) and (3.6). The main result in this three-dimensional setting is contained in the following theorem. Theorem 3.5 ([36, 43, 44]). To any initial data 1 uin ; E in ; B in 2 HP 2 R3 ; there corresponds an existence time T > 0 and a unique local solution of (3.8),

1 3 u 2 L1 .0; T /I HP 2 \ L2 .0; T /I HP 2 \ L1 ;

1 1 E 2 L1 .0; T /I HP 2 \ L2 .0; T /I HP 2 ;

1 1 3 B 2 L1 .0; T /I HP 2 \ L2 .0; T /I HP 2 C HP 2 : Furthermore, the solution is global (i.e., T D 1) if the initial data are sufficiently small.

3.2.3 Weak-strong stability and dissipative solutions On the one hand, as already explained, there is no known global well-posedness theory for the systems (3.5), (3.6) and (3.8) in the energy space, due to their lack of weak stability. On the other hand, in Sections 3.2.1 and 3.2.2, we have briefly presented theorems on the existence and uniqueness of strong solutions to the system (3.8). Around such smooth solutions and in order to circumvent the lack of weak stability, we introduce now the dissipative solutions of these incompressible Navier– Stokes–Maxwell systems. Generally speaking, the concept of dissipative solutions is based on the weak-strong stability, when available, of a given system, i.e., the uniqueness of all weak solutions provided at least one strong solution exists. It seems that such weak-strong stability principles were first introduced by Dafermos [24] in the context of conservation laws. Dissipative solutions are not new in fluid and gas dynamics. They are precisely employed to treat the instability of nonlinear terms in the energy space. Lions first


85

defined them for the Boltzmann equation in [54]. He then established their existence for the incompressible Euler system in [59, Section 4.4], as an alternative to the very weak notion of measure-valued solutions introduced by DiPerna and Majda [34], which have later been shown in [16] to be actually stronger (at least not weaker, as each measure-valued solution is shown to be a dissipative solution, as well). It can more easily be shown that any weak solution of the incompressible Euler system is a dissipative solution (see [25, Appendix B] for a proof). This, however, is not known to hold in general for renormalized solutions of the Boltzmann equation, i.e., renormalized solutions are not known, in general, to be dissipative solutions as defined by Lions in [54]. It is sometimes argued that dissipative solutions are too weak and that they do not express any physical reality, because they are not shown to be unique in general. Even so, they do enjoy certain definite qualities: they exist globally in time for large initial data in the energy space; they coincide with the unique strong solution when the latter exists; they allow energy dissipation phenomena to occur. The last property above is especially significant in light of recent results on the energy dissipation in the incompressible Euler flow establishing, in particular, the existence of weak solutions with kinetic energy strictly decaying (or increasing, which is equivalent since the Euler flow is reversible) over time (see [25, 72]). This energy dissipation cannot hold beyond a certain regularity threshold (see [22, 23] on Onsager’s conjecture) and, therefore, it is crucial to consider rather low regularity weak solutions of the incompressible Euler system in order to understand energy dissipation and turbulent flow. In this context, we wish to mention the striking recent developments [17, 18, 19, 26, 27] demonstrating the existence of energy-dissipating flows enjoying some Hölder regularity. Dissipative solutions have found an important application in a wide range of asymptotic problems, for they are especially well adapted, through relative entropy methods (or modulated energy methods), to situations presenting a lack of compactness. In particular, they were employed by the second author in [69, 71] to establish the hydrodynamic convergence of renormalized solutions of the Boltzmann equation towards dissipative solutions of the incompressible Euler system (see also [70]). Another application by Brenier [15] concerns the convergence of the Vlasov–Poisson system towards the incompressible Euler equations in the quasi-neutral regime. 3.2.3.1 The incompressible Navier–Stokes–Maxwell system Let us explain now how the energy (3.9) can be modulated and establish a weak-strong stability principle, which will eventually lead to a suitable notion of dissipative solution for the incompressible Navier–Stokes–Maxwell system (3.8) in any dimension. We will then move on to apply the same strategy to the more complex systems (3.5) and (3.6) by modulating the energies from Proposition 3.3 and thus produce similar dissipative solutions.

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Proposition 3.6. Let .u; E; B/ be a smooth solution to the incompressible Navier– N BN 2 Stokes–Maxwell system (3.8). Further, consider test functions u; N jN; E; Cc1 .Œ0; 1/ R3 / such that (

@t EN rot BN D jN; div uN D 0; div BN D 0: @t BN C rot EN D 0;

(3.12)

We define the acceleration operator by N C x uN C P jN ^ BN N BN D @t uN P .uN 1rx u/ ; A u; N jN; E; jN C EN C uN ^ BN and the growth rate by

.t/ D

2C02 2 2 jN.t/2 3 ; C C ku.t/k N L1 Lx x

where C0 > 0 denotes the operator norm of the Sobolev embedding HP 1 .R3 / ,! L6 .R3 /. Then, one has the stability inequality ıE .t/ C

1 2

Z

t

0 Rt

ıE .0/e

0

ıD .s/e

Rt s

. /d

ds

Z t Z .s/ds

C

A

0

R3

Rt u uN dx .s/e s . /d ds; N j j

(3.13)

where the modulated energy ıE and energy dissipation ıD are given by 2 2 1 1 1 N .t/k2L2 C E EN .t/L2 C B BN .t/L2 ; k.u u/ x x x 2 2 2 (3.14) 1 2 2 N N C j j .t/ L2 : ıD .t/ D krx .u u/.t/k L2 x x ıE .t/ D

Proof. We have already established formally in (3.9) the conservation of the energy for the system (3.8). The very same computations applied to the test functions N BN yield the identity u; N jN ; E; Z 2 2 2 1 1 d uN 2 2 N N N L2 C E L2 C B L2 C krx uk N L2 C A N dx: jN L2 D kuk x x x x x j 2 dt R3


87

Furthermore, another similar duality computation gives Z Z d 2 N N N 2rx u W rx uN C j j dx u uN C E E C B B dx C dt R3 R3 Z D uN ˝ .u u/ N W rx .u u/ N dx 3 ZR

N uN C .u u/ N jN dx C .j jN / ^ .B B/ N ^ .B B/ 3 ZR u A dx: j R3 On the whole, combining the above identities with the formal energy conservation (3.9), we find 2 2 2 1 1 d N 2L2 C E EN L2 C B BN L2 C krx .u u/k N 2L2 C j jN L2 ku uk x x x x x 2 dt Z D uN ˝ .u u/ N W rx .u u/ N dx R3 Z

N .u u/ N uN dx jN ^ .B B/ N .j jN/ ^ .B B/ C 3 ZR u uN C A dx: j jN R3 The next step consists in estimating the terms in the right-hand side above that are nonlinear in .u; j; E; B/ and to absorb the resulting expressions with the modulated energy ıE .t/ and the modulated energy dissipation ıD .t/ by suitable uses of Young’s inequality and Grönwall’s lemma. Thus, we obtain d ıE .t/ C ıD .t/ dt N L2x krx .u u/k N L2x kuk N L1 ku uk x B BN 2 j jN 2 N L6x C kuk C jN L3 B BN L2 ku uk N L1 Lx Lx x x x Z u uN C A dx j jN R3 2 C02 1 2 2 2 B BN 2 2 N N L1 uk N uk N kuk j k ku 1 C 3 2 C L Lx Lx Lx x x 2 Z 1 u uN j jN 2 2 C C krx .u u/k N 2L2 C A dx: L x x j jN 2 2 R3

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Hence, 1 d ıE .t/ C ıD .t/ .t/ıE .t/ C dt 2

u uN A dx; j jN R3

Z

which concludes the proof of the proposition with a direct application of Grönwall’s lemma. Note that the test functions satisfying the linear constraints (3.12) are easily constructed by considering scalar potentials 'N 2 Cc1 .Œ0; 1/ R3 / and vector potentials AN 2 Cc1 .Œ0; 1/ R3 / and then setting EN D rx 'N @t AN

and

N BN D rot A:

(3.15)

One may prefer, for various reasons, to deal, in a completely equivalent manner, with N BN 2 Cc1 .Œ0; 1/ R3 / satisfying the stationary constraints test functions u; N jN; E; div uN D 0; div BN D 0; jN D EN C uN ^ BN ; rather than the constraints (3.12). In this case, instead of (3.13), we obtain the stability inequality Z Rt 1 t ıE .t/ C ıD .s/e s . /d ds 2 0 0 1 3 2 Z t Z u uN Rt Rt 4 ıE .0/e 0 .s/ds C A @E EN A dx 5 .s/e s . /d ds; R3 0 B BN where the acceleration operator is now defined by 1 0 N C x uN C P jN ^ BN @t uN P .uN rx u/ N BN D @ A: A u; N jN; E; @t EN C rot BN jN @t BN rot EN The preceding proposition provides an important weak-strong stability property for the incompressible Navier–Stokes–Maxwell system (3.8). Indeed, the stability N B/ N of (3.8) such that inequality (3.13) essentially implies that a solution .u; N jN; E; 2 3 N and j 2 L L , if it exists, is unique in the whole class of weak solutions uN 2 L2t L1 x t x in the energy space, for any given initial data. Remark. In order to impose minimal local integrability assumptions on the test function u, N it is tempting to employ the method of Lions and Masmoudi [58] for estimating the nonlinear term ŒuN ˝ .u u/ N W rx .u u/ N by splitting uN D u1 N fjujKg C N , for some large K > 0, which yields u1 N fjuj>Kg N N L1x N W rx .u u/k kŒuN ˝ .u u/

3 ku uk N L6x krx .u u/k K ku uk N L2x krx .u u/k N L2x C u1 N fjuj>Kg N L2x N L x

1 K2 3 krx .u u/k N 2L2 C krx .u u/k N 2L2 C C0 u1 N 2L2 ; N fjuj>Kg ku uk N Lx x x x 4


89

for any > 0, where C0 > 0 denotes the operator norm of the Sobolev embedding N fjuj>Kg is arbitrarily HP x1 ,! L6x . Then, choosing K > 0 large enough so that u1 3 N Lx

small and setting > 0 small enough, it is readily seen that the last two terms above can be absorbed by the modulated entropy dissipation. Of course, the choice of the parameter K is not uniform for all uN 2 L3x . This approach definitely allows us 1 3 to merely consider velocity fields uN 2 L2t L1 x C Lt Lx when establishing weakstrong stability principles for the incompressible Navier–Stokes system (see [58]). Here, however, considering the coupling of the fluid equations with Maxwell’s system introduces other nonlinear terms in the estimates, which unfortunately require that uN 2 L2t L1 x in order to be duly controlled. By analogy with Lions’ dissipative solutions to the incompressible Euler system [59, Section 4.4], we provide now a suitable notion of dissipative solution for the incompressible Navier–Stokes–Maxwell system (3.8), based on Proposition 3.6, and establish their existence next. Definition. We say that .u; E; B/ 2 L1 Œ0; 1/I L2 R3 \ C Œ0; 1/I w-L2 R3 ; such that

div u D 0;

div B D 0;

is a dissipative solution of the incompressible Navier–Stokes–Maxwell system (3.8), if it solves in the sense of distributions the Maxwell equations @t E rot B D j; @t B C rot E D 0; with the Ohm law

j D .E C u ^ B/ ; N BN 2 Cc1 .Œ0; 1/ R3 / satisfying the linear and if, for any test functions u; N jN; E; constraints (3.12), the stability inequality (3.13) is verified. As previously mentioned, dissipative solutions define actual solutions in the sense that they coincide with the unique strong solution when the latter exists. The following theorem asserts their existence. Theorem 3.7. For any initial data uin ; E in ; B in 2 L2 .R3 / such that div uin D 0;

div B in D 0;

there exists a dissipative solution to the incompressible Navier–Stokes–Maxwell system (3.8).

90


Proof. Following Lions [59], we easily build the dissipative solutions by introducing viscous approximations of the system (3.8). Thus, for each > 0, we consider weak solutions of the system 8 ˆ < @t u C u rx u x u D rx p C j ^ B ; div u D 0; @t E rot B D j ; j D .E C u ^ B /; ˆ : div B D 0; @t B C rot E x B D 0; (3.16) associated with the initial data uin ; E in ; B in and satisfying for all t > 0 the energy inequality

1 ku k2L2 C kE k2L2 C kB k2L2 .t/ x x x 2 Z t 1 krx u .s/k2L2 C kj .s/k2L2 C krx B .s/k2L2 ds C x x x 0

1 uin 2 2 C E in 2 2 C B in 2 2 : Lx Lx Lx 2 Such weak solutions are easily established following the method of Leray [50], because the nonlinear term j ^ B is now stable with respect to weak convergence in the energy space defined by the above energy inequality, thanks to the dissipation on B . Then, repeating the computations of Proposition 3.6, it is readily seen that Z Z 2 d 2rx u W rx uN C j jN dx u uN C E EN C B BN dx C dt R3 R3 Z D uN ˝ .u u/ N W rx .u u/ N dx R3 Z

N uN C .u u/ N jN dx C N ^ .B B/ .j jN/ ^ .B B/ 3 ZR Z u A dx rx B W rx BN dx: j R3 R3 Hence, defining the modulated energy ıE .t/ and modulated energy dissipation ıD .t/ by simply replacing .u; j; E; B/ by .u ; j ; E ; B / in (3.14), we infer that Z t

ıD .s/ C krx B .s/k2L2 ds ıE .t/ C x 0 Z tZ uN ˝ .u u/ N W rx .u u/ N dxds ıE .0/ C R3 0 Z tZ

N .u u/ N uN dxds N .j jN/ ^ .B B/ jN ^ .B B/ C R3 0 Z tZ u uN A N C rx B W rx BN dxds: C j j R3 0


91

Then, following the proof of Proposition 3.6, we arrive at Z t 1 2 ıE .t/ C ıD .s/ C krx B .s/kL2 ds x 2 0 Z Z t u uN ıE .0/ C C rx B W rx BN dx ds

.s/ıE .s/ C A j jN 0 R3 Z Z t u uN dx ds

.s/ıE .s/ C A ıE .0/ C j jN R3 0 Z t 2 C krx B k2L2 C rx BN L2 ds; x x 2 2 0 and an application of Grönwall’s lemma yields Z Rt Rt 1 t ıD .s/e s . /d ds ıE .0/e 0 .s/ds ıE .t/ C 2 0 Z t Z Rt 2 u uN C A N dx C rx BN L2 .s/e s . /d ds: x j j 2 R3 0 We may now pass to the limit in the above stability inequality. Thus, up to extraction of subsequences, we may assume that, as ! 0,

u * u

2 2 P1 in L1 t Lx \ Lt Hx ;

j * j

in L2t L2x ;

E * E

2 in L1 t Lx ;

B * B

2 in L1 t Lx :

Furthermore, noticing that @t u , @t E and @t B are uniformly bounded, in L1loc in time and in some negative index Sobolev space in x, it is possible to show (see [59, Appendix C]) that .u ; E ; B / converges to .u; E; B/ 2 C.Œ0; 1/I w-L2 .R3 // weakly in L2x , uniformly locally in time. Then, by the weak lower semi-continuity of the norms, we obtain that, for every t > 0, Z Z Rt Rt 1 t 1 t ıE .t/ C ıD .s/e s . /d ds lim inf ıE .t/ C ıD .s/e s . /d ds: !0 2 0 2 0 Hence, the stability inequality (3.13) holds. Finally, using again that @t u is uniformly bounded, in L1loc in time and in some negative index Sobolev space in x (in fact, we see from (3.16) that it is bounded in L2t L1x C Hx1 ), and that u is uniformly bounded in L2t HP x1 , we infer, invoking a classical compactness result by Aubin and Lions [9, 52] (see [73] for a sharp compactness criterion; see also [49, Section 12.1]), that the u ’s converge to u strongly in L2loc .dtdx/. Therefore, it is readily seen that Ohm’s law is satisfied asymptotically, which concludes the proof of the theorem.

92


3.2.3.2 The two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law Following the strategy of Proposition 3.6, the next result establishes a crucial weak-strong stability principle for the two-fluid incompressible Navier–Stokes– Maxwell system with Ohm’s law (3.5). Proposition 3.8. Let .u; n; E; B/ be a smooth solution to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5). Further, consider test N BN 2 Cc1 .Œ0; 1/ R3 / such that functions u; N n; N jN; E; 8 div uN D 0; ˆ < N N N N @t E rot B D j ; div EN D n; (3.17) ˆ : N @t B C rot EN D 0; div BN D 0: We define the acceleration operator by N C N C 12 P nN EN C jN ^ BN xu N BN D @t uN 1 P .uN rxu/ ; A u; N n; N jN; E; 2 jN nN uN C 12 12 rx nN C EN C uN ^ BN and the growth rate by 3C02 3 4 2

.t/ D C C 2 ku.t/k N 1C Lx

! 2 1 2 1 rx nN EN .t/ C jN.t/ 3 ; 3 2 2 Lx Lx

where C0 > 0 denotes the operator norm of the Sobolev embedding HP 1 .R3 / ,! L6 .R3 /. Then, one has the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 Z t Z Rt Rt u uN A dx .s/e s . /d ds; ıE .0/e 0 .s/ds C N j j n.u u/ N R3 0 (3.18) where the modulated energy ıE and energy dissipation ıD are given by 1 1 N .t/k2L2 C k.n n/ N .t/k2L2 k.u u/ x x 2 8 2 1 1 2 C E EN .t/L2 C B BN .t/L2 ; x x 4 4 2 1 2 j nu .jN nN u/ N C N .t/L2 : ıD .t/ D krx .u u/.t/k L2 x x 2 ıE .t/ D

(3.19)

Proof. We have already formally established in Proposition 3.3 the conservation of the energy for systems (3.5). The very same computations applied to the test functions N B/ N yield the identity .u; N n; N jN; E; Z d N uN E .t/ C DN .t/ D A N dx; (3.20) j nN uN dt R3


93

where the energy EN and energy dissipation DN are obtained simply by replacing the unknowns by the test functions in the respective definitions of Proposition 3.3. Furthermore, other similar duality computations yield that Z Z d 1 1 1 N N u uN C nnN C E E C B B dx C 2rx u W rx uN dx dt R3 4 2 2 R3 Z D uN ˝ .u u/ N W rx .u u/ N dx R3 Z

1 .nE C j ^ B/ uN C nN EN C jN ^ BN u dx C 2 R3 Z Z 1 1 1 u N N j E rx nN C j E rx n dx A dx; 0 2 R3 2 2 R3 and 1 .j nu/ .jN nN u/ N 1 1 .j nu C .n n/ N u/ N .jN nN u/ N C .n n/ N uN j nu .jN nN u/ N D 2 2 1 .j nu/ .jN nu/ N C 2 1 1 0 N N N u/ N rx nN C E C uN ^ B A D .j nu C .n n/ j nu C .n n/ N uN 2 2 1 1 1 C N .n n/ N uN j nu .jN nN u/ N C rx n C E C u ^ B .jN nu/; 2 2 2 whence, considering the sum of the preceding relations, Z 1 1 1 d u uN C nnN C E EN C B BN dx dt R3 4 2 2 Z 1 N 2rx u W rx uN C .j nu/ .j nN u/ C N dx R3 Z Z 1 1 N uN ˝ .u u/ N W rx .u u/ N dx C .n n/.u N u/ N D rx nN E dx 2 R3 2 R3 Z

1 N uN C .u u/ N jN dx .j nu .jN nN u// N ^ .B B/ N ^ .B B/ C 2 R3 Z Z 1 u N .n n/.j N nu .j nN u// N uN dx A dx: C j nu C .n n/ N uN 2 R3 R3 On the whole, combining the above identities with the energy decay imposed by the formal energy conservations from Proposition 3.3, we find the following modu-

94


lated energy inequality: d ıE .t/ C ıD .t/ dt Z

Z 1 1 uN ˝ .u u/ N W rx .u u/ N dx .n n/.u N u/ N rx nN EN dx 2 R3 2 R3 Z

1 N uN C .u u/ N jN dx .j nu .jN nN u// N ^ .B B/ N ^ .B B/ 2 R3 Z Z 1 u uN N .n n/.j N nu .j nN u// N uN dx C A dx: j jN n.u u/ N 2 R3 R3

The next step consists in estimating the terms in the right-hand side above that are nonlinear in .u; n; j; E; B/ and absorbing the resulting expressions with the modulated energy ıE .t/ and the modulated energy dissipation ıD .t/ by suitable uses of Young’s inequality and Grönwall’s lemma. Thus, we obtain d ıE .t/ C ıD .t/ dt

1 1 N N L2x krx .u u/k N L2x ku uk N L6x N L2x C rx nN E kuk N L1 ku uk 3 kn nk x 2 2 Lx 1 N L1 N L2x C B BN L2 j nu .jN nN u/ C N L2 kuk kn nk x x x 2 Z 1 u u N N L6x C C jNL3 B BN L2 ku uk A dx x x j jN n.u u/ N 2 R3 ! 2 3 1 1 3C02 2 2 2 N L1 N L1 N 2L2 N L2 C rx nN EN C kuk kuk kn nk ku uk x x x x 2 8 2 3 2 Lx 3C02 jN2 3 B BN 2 2 N 2L1 C C kuk L Lx x x 2 8 2 1 j nu .jN nN u/ C krx .u u/k N 2L2 C N L2 x x 4 Z2 u uN C A dx: j jN n.u u/ N R3

Hence, d 1 ıE .t/ C ıD .t/ .t/ıE .t/ C dt 2

u uN A dx; j jN n.u u/ N R3

Z

which concludes the proof of the proposition with a direct application of Grönwall’s lemma. Again, note that the test functions satisfying the linear constraints (3.17) are easily constructed employing the relations (3.15). Now, to deal, in a com one may prefer N BN 2 Cc1 .Œ0; 1/ R3 / pletely equivalent manner, with test functions u; N n; N jN; E;


95

satisfying the stationary constraints div uN D 0;

div EN D n; N

div BN D 0;

1 jN nN uN D rx nN C EN C uN ^ BN ; 2

rather than the constraints (3.17). In this case, instead of (3.18), we obtain the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 1 3 0 2 Z t Z u uN Rt Rt 4 A @E 1 rx n EN 1 rx nN A dx5.s/e s . /d ds; ıE .0/e 0 .s/ds C 2 2 R3 0 B BN where the acceleration operator is now defined by 1 0 N C x uN C 12 P nN EN C jN ^ BN @t uN P .uN rx u/ 1 N BN D @ A: A u; N n; N jN; E; @t EN C rot BN jN 2 1 N N @t B rot E 2 The preceding proposition provides an important weak-strong stability property for the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5). Indeed, the stability inequality (3.18) essentially implies that a solution 1 2 3 N B/ N of (3.5) such that uN 2 L2t L1 N .u; N n; N jN; E; N EN 2 L2t L3x , if x , j 2 Lt Lx and 2 rx n it exists, is unique in the whole class of weak solutions in the energy space, for any given initial data. By analogy with Lions’ dissipative solutions to the incompressible Euler system [59, Section 4.4], we provide now a suitable notion of dissipative solution for the twofluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), based on Proposition 3.8, and establish their existence next. Definition. We say that

.u; n; E; B/ 2 L1 Œ0; 1/I L2 R3 \ C Œ0; 1/I w-L2 R3 ;

such that

div u D 0; div E D n; div B D 0; is a dissipative solution of the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), if it solves in the sense of distributions the Maxwell equations @t E rot B D j; @t B C rot E D 0;

with the Ohm law

1 j nu D rx n C E C u ^ B ; 2

96


N BN 2 Cc1 .Œ0; 1/ R3/ satisfying the linear and if, for any test functions u; N n; N jN; E; constraints (3.17), the stability inequality (3.18) is verified. As previously mentioned, dissipative solutions define actual solutions in the sense that they coincide with the unique strong solution when the latter exists. The following theorem asserts their existence. Theorem 3.9. For any initial data uin ; nin ; E in ; B in 2 L2 .R3 / such that div uin D 0;

div E in D nin ;

div B in D 0;

there exists a dissipative solution to the two-fluid incompressible Navier–Stokes– Maxwell system with Ohm’s law (3.5). Proof. As in the proof of Theorem 3.7, it is possible, here, to justify the existence of dissipative solutions by introducing viscous approximations of the system (3.5). Thus, for each > 0, we consider weak solutions of the system 8 1 ˆ ˆ @t u C u rx u x u D rx p C .n E C j ^ B / ; div u D 0; ˆ ˆ 2 ˆ ˆ ˆ < div E D n ; @t E rot B x E D j ; @t B C rot E x B D 0; div B D 0; ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ j n u D rx n C E C u ^ B ; : 2 (3.21) associated with the initial data uin ; nin ; E in ; B in and satisfying for all t > 0 the energy inequality 1 1 1 1 2 2 2 2 ku kL2 C kn kL2 C kE kL2 C kB kL2 .t/ x x x x 2 8 4 4 Z t 1 C krx u k2L2 C kj n u k2L2 x x 2 0 2 2 2 C krx n kL2 C krx E kL2 C krx B kL2 .s/ ds x x x 4 2 2 1 2 1 2 1 2 1 2 uin L2 C nin L2 C E in L2 C B in L2 : x x x x 2 8 4 4 Such weak solutions are easily established following the method of Leray [50], for the nonlinear terms n E and j ^ B are now stable with respect to weak convergence in the energy space defined by the above energy inequality, thanks to the dissipation on n , E and B .


97

Then, repeating the computations of Proposition 3.8, it is readily seen that Z d 1 1 1 N N u uN C n nN C E E C B B dx dt R3 4 2 2 Z 1 N 2rx u W rx uN C .j n u / .j nN u/ C N dx R3 Z Z 1 1 D uN ˝ .u u/ N W rx .u u/ N dx C .n n/.u N u/ N rx nN EN dx 2 R3 2 R3 Z

1 N N N N N .u u/^.B N C .j n u .jN nN u//^.B B/ uC B/ j dx 2 R3 Z 1 .n n/.j N n u .jN nN u// N uN dx C 2 R3 Z u A dx j n u C .n n/ N uN R3 Z

rx n W rx nN C rx E W rx EN C rx B W rx BN dx: 2 2 R3 4 Hence, defining the modulated energy ıE .t/ and modulated energy dissipation ıD .t/ by simply replacing .u; n; j; E; B/ by .u ; n ; j ; E ; B / in (3.19), we infer that Z t 1 ıD .s/ C ıE .t/ C krx n k2L2 C krx E k2L2 C krx B k2L2 .s/ ds x x x 2 2 0 Z tZ ıE .0/ C uN ˝ .u u/ N W rx .u u/ N dxds R3 0 Z tZ 1 1 N .n n/.u N u/ N rx nN E dxds 2 0 R3 2 Z Z 1 t N uN dxds N ^ .B B/ .j n u .jN nN u// 2 0 R3 Z Z 1 t N jN dxds N ^ .B B/ .u u/ 2 0 R3 Z tZ 1 .n n/.j N n u .jN nN u// N uN dxds 2 0 R3 Z tZ u uN dxds A C N j jN n .u u/ R3 0 Z tZ

C rx n W rx nN C rx E W rx EN C rx B W rx BN dxds: 2 2 R3 4 0

98


Then, following the proof of Proposition 3.8, we arrive at Z t 1 1 2 2 2 ıE .t/ C ıD .s/ C krx n kL2 C krx E kL2 C krx B kL2 .s/ ds x x x 2 2 2 0 Z Z t u uN ıE .0/ C dx ds

.s/ıE .s/ C A N n .u u/ j N j 3 R 0 Z Z t 1 N N C rx n W rx nN C rx E W rx E C rx B W rx B dxds 2 0 R3 2 Z Z t u uN dx ds

.s/ıE .s/ C A ıE .0/ C N j jN n .u u/ R3 0 Z t 1 2 2 2 C n C E C B krx kL2 krx kL2 krx kL2 ds x x x 4 0 2 Z t 2 2 1 C N 2L2 C rx EN L2 C rx BN L2 ds; krx nk x x x 4 0 2 and an application of Grönwall’s lemma yields Z Rt Rt 1 t ıD .s/e s . /d ds ıE .0/e 0 .s/ds ıE .t/ C 2 0 Z t Z Rt u uN s . /d ds dx .s/e C A N j jN n .u u/ R3 0 Z Rt 2 2 t 1 2 N N C N L2 C rx E L2 C rx B L2 .s/e s . /d ds: krx nk x x x 4 0 2 We may now pass to the limit in the above stability inequality. Thus, up to extraction of subsequences, we may assume that, as ! 0,

u * u


n * n

2 in L1 t Lx ;

E * E

2 in L1 t Lx ;

B * B

2 in L1 t Lx :

Furthermore, noticing that @t u , @t n , @t E and @t B are uniformly bounded, in L1loc in time and in some negative index Sobolev space in x, it is possible to show (see [59, Appendix C]) that .u ; n ; E ; B / converges to .u; n; E; B/ 2 C.Œ0; 1/I w-L2 .R3 // weakly in L2x uniformly locally in time. Moreover, further combining the bound on @t u with the fact that u is uniformly bounded in L2t HP x1 and then invoking a classical compactness result by Aubin and Lions [9, 52] (see [73] for a sharp compactness


99

criterion; see also [49, Section 12.1]), we infer that the u ’s converge to u strongly in L2loc .dtdx/. In particular, it follows that, up to extraction of subsequences,

n u * nu

1 2 6 in L1 t Lx \ Lt Lx ;

j n u * j nu in L2t L2x : Then, by the weak lower semi-continuity of the norms, we obtain that, for every t > 0, Z Z Rt Rt 1 t 1 t . /d s ıE .t/ C ıD .s/e ds lim inf ıE .t/ C ıD .s/e s . /d ds: !0 2 0 2 0 Hence, the stability inequality (3.18) holds. Finally, it is readily seen that Ohm’s law is satisfied asymptotically, which concludes the proof of the theorem. We present now an alternative kind of stability inequality for the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), whose understanding will be crucial for the relative entropy method – developed later on in Chapter 12 – in the hydrodynamic limit of the two-species Vlasov–Maxwell–Boltzmann system (2.36). It is based on the identity (3.7) linking the Lorentz force with the Poynting vector E ^ B, which will allow us to stabilize the modulated nonlinear terms solely with the modulated energy ıE (i.e., without absorbing nonlinear terms with the modulated dissipation ıD ; note the different coefficient in front of ıD in the stability inequalities (3.18) and (3.22) below). Proposition 3.10. Let .u; n; E; B/ be a smooth solution to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5). Further, consider test N B/ N 2 Cc1 .Œ0; 1/ R3 / such that kuk functions .u; N n; N jN ; E; N L1 < 1 and t;x 8 div uN D 0; ˆ < N @t EN rot BN D jN ; div EN D n; ˆ : N @t B C rot EN D 0; div BN D 0: We define the acceleration operator by N C N C 12 P nN EN C jN ^ BN xu N BN D @t uN 1 P .uN rxu/ ; A u; N n; N jN ; E; 2 jN nN uN C 12 12 rx nN C EN C uN ^ BN and the growth rate by

p 2 jN nN uN .t/L1 x

.t/ D C 1 1 ku.t/k N 1 2 1 u.t/k N k Lx Lx 1 N N C 2 rx nN E uN ^ B .t/ 1 : 2 krt;x u.t/k N L1 x

Lx

100


Then, one has the stability inequality Z t Rt ıD .s/e s . /d ds ıE .t/ C 0 Z t Z Rt Rt u uN .s/ds 0 s . /d ds; C A dx .s/e ıE .0/e j nu .jN nN u/ N R3 0 (3.22) where the modulated energy ıE and energy dissipation ıD are given by 1 1 N .t/k2L2 C k.n n/ N .t/k2L2 k.u u/ x x 2 8 2 2 1 1 C E EN .t/L2 C B BN .t/L2 x x 4Z 4 1 E EN .t/ ^ B BN .t/ u.t/ N dx; 2 R3 2 1 2 j nu .jN nN u/ N C ıD .t/ D krx .u u/.t/k N .t/L2 : L2 x x 2 ıE .t/ D

(3.23)

Proof. Following the proof of Proposition 3.8, using that div u D div uN D 0, we consider first the identity Z Z d 1 1 1 u uN C nnN C E EN C B BN dx C 2rx u W rx uN dx dt R3 4 2 2 R3 Z D .u u/ N ˝ .u u/ N W rx uN dx R3 Z

1 C .nE C j ^ B/ uN C nN EN C jN ^ BN u dx 2 R3 Z Z 1 1 1 u j EN rx nN C jN E rx n dx A dx 0 3 2 R3 2 2 R Z D .u u/ N ˝ .u u/ N W rx uN dx R3 Z 1 C .n n/ N E EN C j jN ^ B BN uN dx 2 R3 Z 1 jN nN uN .u u/ N ^ B BN dx C 2 R3 Z 1 1 C .n n/ N .u u/ N rx nN EN uN ^ BN dx 2 R3 2 Z 1 1 .j nu/ C rx nN EN uN ^ BN 2 R3 2 1 N C j nN uN rx n E u ^ B dx 2 Z u A dx: 0 R3


Further, using Ohm’s law we obtain Z 1 d 1 1 u uN C nnN C E EN C B BN dx dt R3 4 2 2 Z 1 N 2rx u W rx uN C .j nu/ j nN uN dx C 3 Z R .u u/ N ˝ .u u/ N W rx uN dx D R3 Z 1 .n n/ N E EN C j jN ^ B BN uN dx C 2 R3 Z 1 jN nN uN .u u/ N ^ B BN dx C 2 R3 Z 1 1 N N C .n n/ N .u u/ N rx nN E uN ^ B dx 2 R3 2 Z u A dx: j nu R3

101

(3.24)

Then, expressing the modulated Lorentz force with a modulated Poynting vector as @t

ˇ ˇ2 ˇ2 ˇ 1 E EN ^ B BN C rx ˇE EN ˇ C ˇB BN ˇ 2 N rx E E ˝ E EN C B BN ˝ B BN D .n n/ N E EN j jN ^ B BN ;

we arrive at the relation Z 1 1 1 d 1 N N N N u uN C nnN C E E C B B C E E ^ B B uN dx dt R3 4 2 2 2 Z 1 E EN ^ B BN @t uN dx 3 2 ZR 1 N C 2rx u W rx uN C .j nu/ j nN uN dx 3 Z R .u u/ N ˝ .u u/ N W rx uN dx D R3 Z 1 E EN ˝ E EN C B BN ˝ B BN W rx uN dx 3 2 R Z 1 jN nN uN .u u/ N ^ B BN dx C 2 R3 Z 1 1 N N C .n n/ N .u u/ N rx nN E uN ^ B dx 2 R3 2 Z u A dx: j nu R3

102


On the whole, combining the preceding identity with the energy conservation law for test functions (3.20) and the energy decay imposed by the formal energy conservations from Proposition 3.3, we obtain the following modulated energy inequality: d ıE .t/ C ıD .t/ dt Z

Z

1 .u u/ N ˝ .u u/ N W rx uN dx E EN ^ B BN @t uN dx R3 R3 2 Z 1 C E EN ˝ E EN C B BN ˝ B BN W rx uN dx 2 R3 Z 1 jN nN uN .u u/ N ^ B BN dx 2 R3 Z 1 1 N N .n n/ N .u u/ N rx nN E uN ^ B dx 2 R3 2 Z u uN C A dx: j nu .jN nN u/ N R3

The next step consists in estimating the terms in the right-hand side above that are nonlinear in .u; n; j; E; B/ and absorbing the resulting expressions with the modulated energy ıE .t/ by suitable uses of Young’s inequality and Grönwall’s lemma. In this way we obtain d ıE .t/ C ıD .t/ dt

2 2 1 1 N L1 C krt;x uk E EN L2 C B BN L2 ku x x x 2 2 ! 1 1 N L2x N L2x ku uk N EN u^ N BN kn nk C jN nN uN L1 B BN L2 C rx n x x 2 2 L1 x Z u uN C A dx j nu .jN nN u/ N R3 2 2 1 1 2 N N uk N krt;x uk N L1 C C E E B B ku L2 L2 L2 x x x x 2 2 p 2 1 2 jN nN uN 1 ku uk C N 2L2 C B BN L2 L x x x 4 2 1 1 1 2 2 N N nk N uk N n N E u N ^ B C C r kn ku x 1 L2 L2 x x 2 2 4 Lx Z u uN A dx: C j nu .jN nN u/ N R3 uk N 2L2 x


103

Hence, further noticing that 2 2 1 1 1 1 2 2 N N N L2 C kn nk N L2 C 1 kuk N L1 E E L2 C B B L2 ku uk x x x x x 2 8 4 4 1 1 1 1 2 2 N 2L2 C kn nk N 2L2 C E EN L2 C B BN L2 ku uk x x x x 2 4 4 Z 8 ˇ ˇˇ ˇ 1 ˇE EN ˇ ˇB BN ˇ dx N L1 kuk x 2 R3 2 2 1 1 1 1 N 2L2 C kn nk N 2L2 C E EN L2 C B BN L2 ku uk x x x x 2 Z 8 4 4 1 E EN ^ B BN uN dx 2 R3 D ıE .t/; we find, since kuk N L1 < 1, that t;x d ıE .t/ C ıD .t/ .t/ıE .t/ C dt

u uN A dx; j nu .jN nN u/ N R3

Z

which concludes the proof of the proposition with a direct application of Grönwall’s lemma. Remark. Notice that the preceding method of modulation of the Poynting vector is not applicable to the incompressible Navier–Stokes–Maxwell system (3.8), because the divergence of the electric field E is not determined therein, i.e., Gauss’ law div E D n cannot be used to provide a bound on div E. The preceding proposition provides another weak-strong stability property for the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5). InN B/ N deed, the stability inequality (3.22) essentially implies that a solution .u; N n; N jN; E; 1 1 N 1 1 1 of (3.5) such that uN 2 L1 , r u N 2 L L , j n N u N 2 L L and uk N < 1, if k t;x Lt;x t;x t x t x it exists, is unique in the whole class of weak solutions in the energy space, for any given initial data. We do not know whether the condition kuk N L1 < 1 in Proposition 3.10 is actually t;x necessary or merely a technical limitation. Nevertheless, this result shows that such a condition has a stabilizing effect on the two-fluid incompressible Navier–Stokes– Maxwell system with Ohm’s law (3.5). Furthermore, this restriction is physically relevant, since it forces the modulus of the bulk velocity uN to remain everywhere and at all times below the speed of light. More precisely, keeping track of the relevant physical constants in the formal derivations of Chapter 2, we see that the system (3.5)

104


can be recast as 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C .cnE C j ^ B/ ; ˆ ˆ ˆ 2 ˆ ˆ ˆ 1 ˆ ˆ div E D cn; @t E rot B D j; < c 1 ˆ ˆ @t B C rot E D 0; div B D 0; ˆ ˆ c ˆ ˆ ˆ ˆ 1 ˆ ˆ j nu D rx n C cE C u ^ B ; : 2 where the constant c > 0 denotes the speed of light. Notice that the formal energy conservation law satisfied by this system is independent of c > 0 and is thus given by Proposition 3.3. Moreover, expressing the Lorentz force with the Poynting vector as in (3.7) yields now 1 1 @t .E ^ B/ C rx E 2 C B 2 rx .E ˝ E C B ˝ B/ D cnE j ^ B: c 2 Therefore, applying the proof of Proposition 3.10 to the preceding system, we arrive at a stability inequality valid under the restriction that the bulk velocity remains bounded by the speed of light kuk N L1 < c, which is natural. t;x Following the previous developments, it is also possible to use now the stability inequality (3.22) from Proposition 3.10 to define another notion of dissipative solutions for the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law (3.5), whose existence is then established by reproducing the arguments from Theorem 3.9. Indeed, applying the computations from the proof of Proposition 3.10 to the viscous approximation (3.21) only produces new dissipative terms which are easily controlled in the limit ! 0 (note that the condition N L1 0 denotes the operator norm of the Sobolev embedding HP 1 .R3 / ,! L6 .R3 /. Then, one has the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 (3.27) Z t Z Rt Rt u uN .s/ds . /d 0 s ıE .0/e C A dx .s/e ds; j jN R3 0 where the modulated energy ıE and the modulated energy dissipation ıD are given by 2 2 1 1 1 N .t/k2L2 C E EN .t/L2 C B BN .t/L2 ; ıE .t/ D k.u u/ x x x 2 4 4 1 2 2 N C j jN .t/L2 : ıD .t/ D krx .u u/.t/k L2 x x 2 Proof. We have already formally established in Proposition 3.3 the conservation of the energy for system (3.6). The very same computations applied to the test functions N B/ N yield the identity .u; N jN; E; Z d N uN N E .t/ C D .t/ D A N dx; (3.28) j dt R3 where the energy EN and energy dissipation DN are obtained simply by replacing the unknowns by the test functions in the respective definitions of Proposition 3.3. Furthermore, another similar duality computation yields that Z Z 1 1 1 d u uN C E EN C B BN dx C 2rx u W rx uN C j jN dx dt R3 2 2 R3 Z uN ˝ .u u/ N W rx .u u/ N dx D R3 Z

1 N uN C .u u/ N jN dx .j jN/ ^ .B B/ N ^ .B B/ C 2 3 Z R u A dx: j R3 (3.29)


107

On the whole, combining the above identities with the energy decay imposed by the formal energy conservations from Proposition 3.3, we arrive at the following modulated energy inequality: d ıE .t/ C ıD .t/ dt Z

uN ˝ .u u/ N W rx .u u/ N dx Z

1 N uN C .u u/ N jN dx .j jN/ ^ .B B/ N ^ .B B/ 2 R3 Z u uN A dx: C j jN R3 R3

The next step consists in estimating the terms in the right-hand side above that are nonlinear in .u; j; E; B/ and absorbing the resulting expressions with the modulated energy ıE .t/ and the modulated energy dissipation ıD .t/ by suitable uses of Young’s inequality and Grönwall’s lemma. In this way we obtain d ıE .t/ C ıD .t/ dt N L2x krx .u u/k N L2x kuk N L1 ku uk x 1 1 B BN 2 j jN 2 N L1 N L6x C kuk C jN L3 B BN L2 ku uk L Lx x x x x 2 Z2 u uN C A dx j jN 3 R 2 2 C02 1 2 2 2 N L1 N L2 C kuk N L1 C kuk jN L3 B BN L2 ku uk x x x x x 2 Z 1 u uN j jN2 2 C C krx .u u/k N 2L2 C A dx: Lx x j jN 3 2 4 R Hence, 1 d ıE .t/ C ıD .t/ .t/ıE .t/ C dt 2

u uN A dx; j jN R3

Z

which concludes the proof of the proposition with a direct application of Grönwall’s lemma. Note that the test functions satisfying the linear constraints (3.26) are easily constructed by considering vector potentials AN 2 Cc1 .Œ0; 1/ R3 / and then setting EN D @t rot AN

and

N BN D rot rot A:

Now, one may prefer to deal, in a completely equivalent manner, with test functions N B/ N 2 Cc1 .Œ0; 1/ R3 / and jN 2 C 1 .Œ0; 1/ R3 / (here, we cannot impose .u; N E;

108


that jN be compactly supported) satisfying the stationary constraints div uN D 0; div EN D 0; div BN D 0; jN D P EN C uN ^ BN ; rather than the constraints (3.26). In this case, instead of (3.27), we obtain the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 0 2 1 3 Z t Z u uN Rt Rt 4 ıE .0/e 0 .s/ds C A @E EN A dx 5 .s/e s . /d ds; R3 0 B BN where the acceleration operator is now defined by 1 0 N C x uN C 12 P jN ^ BN @t uN P .uN rx u/ 1 N BN D @ N N A: N A u; N jN; E; @ 2 t E C rot B j 1 N N @t B rot E 2 The preceding proposition provides an important weak-strong stability property for the two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6). Indeed, the stability inequality (3.27) essentially implies that a 2 3 N B/ N of (3.6) such that uN 2 L2t L1 N solution .u; N jN; E; x and j 2 Lt Lx , if it exists, is unique in the whole class of weak solutions in the energy space, for any given initial data. By analogy with Lions’ dissipative solutions to the incompressible Euler system [59, Section 4.4], we provide now a suitable notion of dissipative solution for the two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6), based on Proposition 3.11, and establish their existence next. Definition. We say that

.u; E; B/ 2 L1 Œ0; 1/I L2 R3 \ C Œ0; 1/I w-L2 R3 ;

such that

div u D 0;

div E D 0;

div B D 0;

is a dissipative solution of the two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6), if it solves the Maxwell equations @t E rot B D j; @t B C rot E D 0; with the solenoidal Ohm law j D .rx pN C E C u ^ B/ ; N B/ N 2 Cc1 .Œ0; 1/ in the sense of distributions, and if, for any test functions .u; N jN; E; 3 R / satisfying the linear constraints (3.26), the stability inequality (3.27) is verified.


109

As previously mentioned, dissipative solutions define actual solutions, in the sense that they coincide with the unique strong solution when the latter exists. The following theorem asserts their existence. Theorem 3.12. For any initial data uin ; E in ; B in 2 L2 .R3 / such that div uin D 0;

div E in D 0;

div B in D 0;

there exists a dissipative solution to the two-fluid incompressible Navier–Stokes– Maxwell system with solenoidal Ohm’s law (3.6). Proof. As in the proof of Theorems 3.7 and 3.9, it is possible, here, to justify the existence of dissipative solutions by introducing viscous approximations of the system (3.6). However, it will be much more judicious to recover the system (3.6) as an asymptotic regime of the two-fluid incompressible Navier–Stokes–Maxwell system (2.53) for very weak interspecies interactions, which we recast here, for all > 0, as 8 C C C @t uC ˆ C u rx u x u ˆ ˆ ˆ ˆ 1 1 ˆ C C ˆ C 2 uC E C uC ˆ u D rx p C ^ B ; div u D 0; ˆ ˆ ˆ ˆ ˆ < @t u C u rx u x u 1 1 ˆ 2 uC div u E C u ˆ u D rx p ^ B ; D 0; ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ div E D 0; @t E rot B D uC ˆ u ; ˆ ˆ ˆ : div B D 0; @t B C rot E D 0; (3.30) in in ; E ; B satisfying associated with initial data u˙in uin D

C uin uCin : 2

The above two-fluid system satisfies, for all t > 0, the energy inequality

1 uC 2 2 C ku k2 2 C kE k2 2 C kB k2 2 .t/ Lx Lx Lx Lx 2 2 # Z t"

1 C u .s/ u .s/ 2 C krx u .s/k2 2 C C rx uC .s/ L2 2 ds Lx x 0 Lx

1 Cin 2 2 2 C E in 2 2 C B in 2 2 : u L2 C uin Lx Lx Lx x 2 Upon further introducing the variables u D

uC C u 2

and

j D

uC u ;

110


the system (3.30) can be rewritten as 8 2 1 ˆ ˆ j rx j x u D rx p C j ^ B ; div u u C u r u C @ ˆ t x ˆ ˆ 4 2 ˆ ˆ 2 ˆ ˆ < .@ j C u r j C j r u j / C 1 j t x x x 2 ˆ ˆ div j ˆ D rx pN C E C u ^ B ; ˆ ˆ ˆ ˆ @t E rot B D j ; div E ˆ ˆ : @t B C rot E D 0; div B

D 0;

D 0; D 0; D 0; (3.31)

and the corresponding energy inequality becomes, for all t > 0,

2 1 1 1 2 2 2 2 ku kL2 C kj kL2 C kE kL2 C kB kL2 .t/ x x x x 2 8 4 4 Z t 2 1 2 2 2 krx u .s/kL2 C C krx j .s/kL2 C kj .s/kL2 ds x x x 4 2 0 2 1 2 j in 2 2 C 1 E in 2 2 C 1 B in 2 2 ; uin L2 C Lx L Lx x x 2 8 4 4 uCin uin

where jin D . Weak solutions of the above systems (3.30) and (3.31) are easily obtained following the method of Leray [50], since the nonlinear terms u˙ ^ B (or, equivalently, u ^ B and j ^ B ) are stable with respect to weak convergence in the energy space defined by the above energy inequalities. N B/ N 2 Cc1 .Œ0; 1/ R3 / satisfying the linear Now, for any test functions .u; N jN ; E; constraints (3.26), we define the approximate acceleration operator by 2 N BN D A u; N BN N jN; E; N jN; E; A u; 4

! P jN rx jN : @t jN C P uN rx jN C jN rx uN x jN

Then, a straightforward energy estimate yields that d dt

2 1 jN2 2 C 1 EN 2 2 C 1 BN 2 2 N 2L2 C kuk Lx Lx Lx x 2 8 4 4 Z 2 2 1 2 uN 2 N N N L2 C A N dx: C krx uk rx j L2 C j L2 D x x x j 4 2 R3


111

Moreover, another similar duality computation gives that Z d 2 1 1 u uN C j jN C E EN C B BN dx dt R3 4 2 2 Z 2 1 N N 2 rx u W rx uN C rx j W rx j C j j dx C 2 R3 Z 2 uN ˝ .u u/ N W rx .u u/ N C uN ˝ j jN W rx .j jN / dx D 4 R3 2 Z

N C jN ˝ .u u/ N W rx .j jN/ dx jN ˝ j jN W rx .u u/ 4 R3 Z

1 N uN C .u u/ N jN dx N ^ .B B/ .j jN/ ^ .B B/ C 2 3 Z R u A dx: j R3 Hence, defining the modulated energy ıE .t/ and the modulated energy dissipation ıD .t/ by ıE .t/ D

1 2 j jN .t/2 2 N .t/k2L2 C k.u u/ Lx x 2 8 2 2 1 1 C E EN .t/L2 C B BN .t/L2 ; x x 4 4

and 2 ıD .t/ D krx .u u/.t/k N C L2 x

2 rx .j jN /.t/2 2 C 1 j jN .t/2 2 ; L Lx x 4 2

respectively, we find that Z t ıD .s/ ds ıE .t/ C 0

ıE .0/ Z tZ C 0 2

R3

Z tZ

2 uN ˝ j jN W rx .j jN/ dxds 4

N C jN ˝ .u u/ N W rx .j jN / dxds jN ˝ j jN W rx .u u/

uN ˝ .u u/ N W rx .u u/ N C

4 0 R3 Z Z

1 t N uN C .u u/ N jN dxds N ^ .B B/ .j jN / ^ .B B/ 2 0 R3 Z tZ u uN A N dxds: C j j R3 0

C

112


The next step consists in estimating the terms in the right-hand side above that are nonlinear in .u ; j ; E ; B / and absorbing the resulting expressions with the modulated energy ıE .t/ and the modulated energy dissipation ıD .t/ by suitable uses of Young’s inequality and Grönwall’s lemma. Thus, we obtain Z t ıE .t/ C ıD .s/ ds 0 Z t N L1 N L2x krx .u u/k N L2x ds ıE .0/ C kuk ku uk x 0 Z 2 t j jN 2 rx .j jN / 2 ds N L1 C kuk L Lx x x 4 0 Z 2 t jN 1 j jN 2 krx .u u/k N L2x C L Lx x 4 0 C jNL3 ku uk N L6x rx .j jN/L2 ds x x Z 1 t 2 j jN 2 ds jN 3 B BN 2 ku uk N 1 B B C N N 6 Ckuk Lx Lx Lx Lx Lx Lx 2 0 Z tZ u uN A N dxds C j j R3 0 ıE .0/ Z t 2 2 C02 1 2 2 2 N N N L1 C B B L2 ds N L2 C kuk N L1 C j L3 kuk ku uk x x x x x 2 0 Z t 2 2 2 2 2 C0 2 2 C C N L2 C C jN L3 krx .uu/k rx .j jN/ L2 ds x x x 2 16 4 8 0 Z t 2 2 1 jN2 1 j jN2 2 N 2L1 C C C kuk Lx Lx x 4 4 4 0 Z u uN C A N dx ds j j R3 Z Z t u uN ıE .0/ C

.s/ıE .s/ C A N dx ds j j R3 0 Z t 1 C C 2 ˇ.s/ ıD .s/ ds; 2 0

where ˇ.t/ D

2 1 C 2 2 jN 1 : N 2L1 C C 02 jNL3 C kuk Lx x x 16 4 2


Hence,

Z

ıE .t/ C

0

t

113

1 ıD .s/ ds 2

ıE .0/ Z Z t C

.s/ıE .s/ C

u uN 2 dx C ˇ.s/ıD .s/ ds; A j jN R3

0

and an application of Grönwall’s lemma yields Z Rt Rt 1 t ıE .t/ C ıD .s/e s . /d ds ıE .0/e 0 .s/ds 2 0 R Z t Z t u uN 2 dx .s/ C ˇ.s/ıD .s/ e s . /d ds: C A N j j 3 R 0 We can now pass to the limit in the above stability inequality. Thus, up to extraction of subsequences, we may assume that, as ! 0,

u * u


j * j

in L2t L2x ;

E * E

2 in L1 t Lx ;

B * B

2 in L1 t Lx :

Furthermore, noticing from (3.31) that @t u , @t E and @t B are uniformly bounded, in L1loc in time and in some negative index Sobolev space in x, one can show (see [59, Appendix C]) that .u ; E ; B / converges to .u; E; B/ 2 C.Œ0; 1/I w-L2 .R3 // weakly in L2x uniformly locally in time. Then, by the weak lower semi-continuity of the norms, we obtain that, for every t > 0, Z Z Rt Rt 1 t 1 t . /d s ıE .t/ C ıD .s/e ds lim inf ıE .t/ C ıD .s/e s . /d ds: !0 2 0 2 0 Hence, further assuming that ıE .0/ ! ıE .0/, as ! 0, the stability inequality (3.27) Cin holds. Notice that the convergence of the initial data is satisfied whenever u uin 2 ! 0, as ! 0. Lx Finally, using again that @t u is uniformly bounded, in L1loc in time and in some negative space in x (in fact, we see from (3.31) that it is bounded index Sobolev in L2t L1x C Hx1 ), and that u is uniformly bounded in L2t HP x1 , we infer, invoking a classical compactness result by Aubin and Lions [9, 52] (see [73] for a sharp compactness criterion; see also [49, Section 12.1]), that the u ’s converge towards u strongly in L2loc dtdx . Therefore, passing to the limit in the evolution equation for j in (3.31), it is readily seen that Ohm’s law is satisfied asymptotically. This concludes the proof of the theorem.

114


As before, we present now an alternative kind of stability inequality for the twofluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6). It is a mere adaptation of Proposition 3.10 to the present case, which relies on the interpretation of the Lorentz force with the Poynting vector. We recall that this method allows us to stabilize the modulated nonlinear terms solely with the modulated energy ıE . Proposition 3.13. Let .u; E; B/ be a smooth solution to the two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6). Further consider N B/ N 2 Cc1 .Œ0; 1/ R3 / such that kuk test functions .u; N jN; E; N L1 < 1 and t;x 8 ˆ
0 denotes the speed of light. Then, applying the proof of Proposition 3.13 to the preceding system, we arrive at a stability inequality valid under the restriction that the bulk velocity remains bounded by the speed of light N L1 < c, which is natural. kuk t;x Following the previous developments, it is also possible to use now the stability inequality (3.32) from Proposition 3.13 to define another notion of dissipative solutions for the two-fluid incompressible Navier–Stokes–Maxwell system with solenoidal Ohm’s law (3.6). The existence of such solutions is then established by reproducing the arguments from Theorem 3.12. The only argument from the proof of Theorem 3.12 that needs special care in order to conclude the existence of dissipative solutions resides in the weak lower semi-continuity of the modulated energy ıE .t/ defined by (3.33), which we have already established in (3.25).

Part II

Conditional convergence results

Chapter 4

Two typical regimes We will now focus on two specific regimes which are critical, in the sense that all the formal asymptotics mentioned in Chapter 2 can be rigorously obtained by similar or even simpler arguments. The first scaling we will investigate here is the one leading from the one-species Vlasov–Maxwell–Boltzmann equations (2.7) to the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (2.33). More precisely, we will set ˛ D , ˇ D and D in (2.7). As discussed in Section 3.1, the resulting limiting model is then very similar to the incompressible Navier–Stokes equations and, thus, the usual methods of hydrodynamic limits will apply. We shall focus specifically on the influence of the electromagnetic field, which introduces numerous technical complications. The second regime we will study is more singular, since the magnetic forcing is much stronger. Specifically, we will consider the scaling leading from the twospecies Vlasov–Maxwell–Boltzmann equations (2.36) to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law (2.80) in the case of strong interspecies collisions, or to the two-fluid incompressible Navier–Stokes– Fourier–Maxwell system with solenoidal Ohm’s law (2.83) in the case of weak interspecies collisions. More precisely, we will set ˛ D ı, ˇ D ı and D 1 in (2.36), with ı= unbounded. Actually, as discussed in Section 3.2, the corresponding limiting models (2.80) and (2.83) are not stable under weak convergence in the energy space, and thus share more similarities with the three-dimensional incompressible Euler equations. So will our proofs of hydrodynamic convergence in this setting. All along this second part on rigorous hydrodynamic convergence proofs, we will consider renormalized solutions of the Vlasov–Maxwell–Boltzmann systems for any number species, whose definition we recall below in Section 4.1. In fact, their existence is not established, which is precisely the reason why the convergence results presented here are deemed conditional, and remains a challenging open problem of outstanding difficulty. Loosely speaking, the specific complexity of the Vlasov–Maxwell–Boltzmann system originates in the nonlinear coupling of the Vlasov–Boltzmann equation with a hyperbolic system, namely, the Maxwell equations. This essential difficulty remains ubiquitous in our analysis of its hydrodynamic limits and is passed on to the most singular asymptotic models present in our work, such as the systems (2.80) and (2.83), whose well-posedness is not fully understood (see Section 3.2) and contains very challenging open questions, as well.

122

4 Two typical regimes

4.1 Renormalized solutions We are now going to recall the notion of renormalized solutions for the Vlasov– Maxwell–Boltzmann systems (2.7) 8 @t f C v rx f C .E C v ^ B/ rv f D Q.f; f /; ˆ ˆ Z ˆ ˆ ˆ ˆ f v dv; @t E rot B D ˆ ˆ ˆ R3 < @t B C rot E D 0; (4.1) ˆ Z ˆ ˆ ˆ ˆ f dv 1; div E D ˆ ˆ ˆ R3 ˆ : div B D 0; and (2.36) 8 ˆ @t f ˙ C v rx f ˙ ˙ .E C v ^ B/ rv f ˙ ˆ ˆ ˆ ˆ ˆ ˆ ˆ @t E rot B ˆ ˆ < @t B C rot E ˆ ˆ ˆ ˆ ˆ div E ˆ ˆ ˆ ˆ ˆ : div B

D Q.f ˙ ; f ˙ / C Q.f ˙ ; f /; Z C D f f v dv; R3

D 0; Z C f f dv; D

(4.2)

R3

D 0;

where we have discarded the free parameters.

4.1.1 The Vlasov–Boltzmann equation Let us focus first on the simpler Vlasov–Boltzmann equation: @t f C v rx f C F rv f D Q .f; f / ; with a given force field F .t; x; v/ satisfying, at least, F; rv F 2 L1loc dtdxI L1 M ˛ dv

(4.3)

for all ˛ > 0:

Recall that M.v/ denotes the global normalized Maxwellian equilibrium M1;0;1 .v/, so that 2 1 ˛ jvj 2 : M˛ D e 3 .2/˛ 2 The above conditions on the force field are minimal requirements so that it is possible to define renormalized solutions of (4.3) (see definition below). We will, however, further restrict the range of applicability of force fields: we assume that rv F D 0, so that the local conservation of mass is verified; we assume that F v D 0, so that the global Maxwellian M.v/ is an equilibrium state of (4.3).

4.1 Renormalized solutions

123

Renormalized solutions of (4.3) are known to exist since the late 1980ies, thanks to DiPerna and Lions [30] (at least for the Boltzmann equation, i.e., for the case F D 0). We are going to briefly describe their derivation and their limitations, and emphasize the main mathematical difficulties preventing their construction for the above Vlasov–Maxwell–Boltzmann systems. Throughout this work, we are interested in the fluctuations of a density f .t; x; v/ around the global normalized Maxwellian M.v/; we will therefore conveniently employ the density G.t; x; v/ defined by f D M G. In this notation, the Vlasov– Boltzmann equation (4.3) reads @t G C v rx G C F rv G D Q .G; G/ ;

(4.4)

where we denote

1 Q.M G; MH /: M Thus, DiPerna and Lions formulated in [30] the first theory yielding global solutions to the Boltzmann equation (4.4), with F D 0, for large initial data G.0; x; v/ D G in .x; v/ 0. Their construction heavily relied on a new notion of solutions, namely, the renormalized solutions. Recall that we utilize the prefixes w- or w - to indicate that a given space is endowed with its weak or weak- topology, respectively.

Q.G; H / D

Definition. We say that a nonlinearity ˇ 2 C 1 .Œ0; 1/I R/ is an admissible renormalization if it satisfies, for some C > 0, ˇ 0 ˇ ˇˇ .z/ˇ

C 1

.1 C z/ 2

for all z 0:

A density function f .t; x; v/ D M G.t; x; v/ 0, where .t; x; v/ 2 Œ0; 1/ R3 R3 , such that G 2 C Œ0; 1/I w-L1loc dxdv \ L1 Œ0; 1/; dtI L1loc dxI L1 .1 C jvj2 /M dv ; (4.5) is a renormalized solution of the Vlasov–Boltzmann equation (4.4) if it solves @t ˇ.G/ C v rx ˇ.G/ C F rv ˇ.G/ D ˇ 0 .G/Q.G; G/

(4.6)

in the sense of distributions for any admissible renormalization, and satisfies for all t > 0 the entropy inequality Z tZ H.f .t// C D.f .s// dxds H.f in / < 1; 0

R3

where f in D M G in is the initial value of f D M G and the relative entropy H.f / D H.f jM / is defined in (1.17), while the entropy dissipation D.f / is defined in (1.13).

124


Note that the renormalized collision operator ˇ 0 .G/Q.G; G/ is well defined in L1loc .dtdxI L1 .M ˛ dv//, with ˛ > 0, for any admissible renormalization, any function in (4.5) and any integrable cross-section b.z; / 2 L1loc .R3 S2 / satisfying the so-called DiPerna–Lions assumption Z 1 b.v v ; / dv d D 0; (4.7) lim jvj!1 jvj2 KS2 for any compact subset K R3 . Indeed, it is possible to deduce directly from (4.7) that (see [4], for instance, for more details), for any ˛ > 0, Z 1 lim b.v v ; /M˛ dv d D 0: jvj!1 jvj2 R3 S2 Therefore, considering first non-negative renormalizations satisfying 0 ˇ 0 .z/ C , the renormalized loss part ˇ 0 .G/Q .G; G/ is easily estimated as 1Cz Z

ˇ 0 .G/Q .G; G/M ˛ dv Z 2 D G .1 C jv j /M

R3

R3

C kGkL1 .1Cjvj2 /Mdv ;

1 1 C jv j2

Z R3 S2

0

ˇ .G/Gb.v v ; /M dvd dv ˛

0 C while the renormalized ˛ gain term ˇ .G/Q .G; G/ is well-defined in the space 1 1 Lloc dtdxI L M dv by the renormalized Vlasov–Boltzmann equation (4.6) because it is the only unestimated expression remaining and it is non-negative. These C controls are easily extended to signed renormalizations satisfying jˇ 0 .z/j 1Cz , for the Vlasov–Boltzmann equation (4.6) is linear with respect to renormalizations so that we may decompose ˇ 0 .z/ with respect to its positive and negative parts. Alternatively and as was originally performed in [30], we could also use the eleG 0G 0 mentary inequality (B.9), setting z D GG 1 and y D log K, with K > 1, which implies that Z Z 0 C ˇ .G/Q .G; G/Mdv K ˇ 0 .G/Q .G; G/M dv R3 R3 0 0 Z 1 G G C MM b dvdv d; ˇ 0 .G/ G 0 G0 GG log log K R3 R3 S2 GG (4.8) to claim that the gain part belongs to L1loc .dtdxI L1 .Mdv//, since it is natural to control the entropy dissipation term above. Finally, it is possible to extend the definition of the renormalized collision operator ˇ 0 .G/Q.G; G/ to all admissible renormalizations by decomposing the renormalized


125

collision integrand as

2 p p ˇ 0 .G/ G 0 G0 GG D ˇ 0 .G/ G 0 G0 GG p

p p C 2ˇ 0 .G/ GG G 0 G0 GG ; and noticing that Z p

f 0 f0

p

2

b.v v ; / dvdv d 0 0 0 0 1 f f b.v v ; / dvdv d D D.f /; f f ff log 4 R3 R3 S2 ff

R3 R3 S2

ff

Z

which follows from the elementary inequality (B.8). Thus, by a solution G of the renormalized equation (4.6), we naturally mean that G should satisfy, for every ˛ > 0 and any non-negative test functions .t; x/ 2 Cc1 Œ0; 1/ R3 and '.v/ 2 W 1;1 .dv/, that Z ˇ G in .0; x/'.v/M ˛ dxdv 3 3 ZR R ˇ .G/ .@t C v rx C F rv / Œ.t; x/'.v/M ˛ dtdxdv Œ0;1/R3 R3 Z D ˇ 0 .G/ Q .G; G/ .t; x/'.v/M ˛ dtdxdv: Œ0;1/R3 R3

The following theorem is a modern formulation of the existence result found in [30]. The existence of renormalized solutions for Vlasov–Boltzmann systems where the force field derives from a self-induced potential, such as the Vlasov–Poisson– Boltzmann system, has been established in [55], while the study of renormalized solutions close to Maxwellian equilibrium has been performed in [56]. Theorem 4.1 ([30, 32]). Let b.z; / be a locally integrable collision kernel satisfying the DiPerna–Lions assumption (4.7) and F .t; x; v/ 2 L1loc .dtdxdv/ a given force field such that 1;1 rv F D 0; F v D0 and F 2 L1loc dtI Wloc .dxdv/ : (4.9) Then, for any initial condition f in D M G in 2 L1loc .dxI L1 ..1 C jvj2 /dv// such that f in D M G in 0 and Z in H.f in / D H.f in jM / D G log G in G in C 1 M dxdv < 1; R3 R3

there exists a renormalized solution f .t; x; v/ D M G.t; x; v/ to the Vlasov– Boltzmann equation (4.4). Moreover, it satisfies the local conservation of mass Z Z f dv C rx f v dv D 0; @t R3

R3

126


and the global entropy inequality, for any t 0, Z tZ D.f .s// dxds H.f in /: H.f .t// C 0

(4.10)

R3

The proof of the above theorem follows the usual steps found in the analysis of weak solutions of partial differential equations, namely, solving an approximate truncated equation, establishing uniform a priori estimates and the weak compactness of the approximate solutions, and finally passing to the limit (by showing the weak stability of nonlinear terms) and, thus, recovering the original equation. It is often the case that these steps reduce to the study of the crucial weak stability of solutions. Thus, for the Vlasov–Boltzmann equation (4.4), the above theorem naturally follows from the weak stability of renormalized solutions, or, in other words, from the weak stability of weak solutions of the renormalized equation (4.6) satisfying the uniform bounds provided by the entropy inequality (4.10). DiPerna and Lions showed the weak stability of the Boltzmann equation, i.e., when F D 0, in [30] and refined their result in [32] by establishing the entropy inequality (4.10). Note that, since we are assuming rv F D 0 and F v D 0, the entropy inequality (4.10) easily follows from formal estimates on the Vlasov– Boltzmann equation (4.4), even when F ¤ 0. Later, Lions improved the method of proof in [53, 54, 55]. We briefly explain now Lions’ strategy, which relies on velocity averaging lemmas, heavy renormalization techniques and, most importantly, on the compactifying (even regularizing, in some cases) effect of the gain term QC .f; f / of the collision operator. To this end, let us consider ˚ a sequence ffk gk2N of actual renormalized solutions to (4.3), with initial data fkin k2N , which converges weakly (at least in L1loc , say) as k ! 1 to f in . We further assume that the initial datum satisfies the following strong entropic convergence lim H fkin D H f in ; k!1

so that the entropy inequality is uniformly satisfied: Z tZ H fk .t/ C D .fk .s// dxds H fkin : 0

(4.11)

R3

Notice that a uniform bound on the entropies H.fk .t// yields, via a direct application inequality (B.3), a uniform bound on fk .t; x; v/ of the elementary Young 1 1 2 in L1 /dv . Moreover, it is possible to show, with a dtI L dxI L .1 C v loc loc slightly more refined application of the Young inequality (B.3) with the Dunford– Pettis compactness criterion (see [68] and Section 5.1 for details), that the fk ’s are in fact weakly relatively compact in L1loc dtdxI L1 .dv/ . Therefore, up to extraction, we may assume that the sequence ffk gk2N converges weakly, as k ! 1, to some f in L1loc dtdxI L1 .dv/ . Similarly, uniform bounds on the nonlinear terms Q˙ .fk ; fk /

R 1 C ıfk v S2 b.; /d

and

ˇ 0 .Gk / Q˙ .fk ; fk / ;


127

where fk D M Gk , for any ı > 0 and any admissible nonlinearity ˇ 2 C 1 .Œ0; 1/I R/, are easily obtained from (4.11) through an estimate similar to (4.8). More precisely, a standard use of the elementary inequality (B.9), setting zD

0 fk0 fk 1 fk fk

and y D log K, with K > 1, yields 0 fk0 fk

0 0 f f 1 0 0 Kfk fk C .f f fk fk / log k k : log K k k fk fk

This functional inequality further implies the weak compactness of the above nonlinear terms, thanks to the Dunford–Pettis compactness criterion (see [68]). Next, we need to improve the convergence of ffk gk2N in t and x by establishing some suitable strong compactness. To this end, we will use some velocity averaging lemma. The main idea is that the transport equation, although it propagates singularities, controls some partial derivatives of the solution: as the bad directions (in which there is no regularity) are related to the velocity variable v, one can prove that, averaging with respect to v, one gets additional regularity in .t; x/. The rather weak version of velocity averaging lemma we use here can be deduced directly from the classical results originally featured in [33] (see Theorems 5 and 6 and the subsequent remarks therein; see also [14, Sections 1.5 and 2.3.1]). It is to be emphasized that the results from [33] are actually stronger than the one presented below, since they yield regularity of the velocity averages. We refer to [5] for a survey of classical velocity averaging results from a modern viewpoint, as well as some interesting open questions in the field. Lemma 4.2 ([33]). Let the bounded family of functions f .t; x; v/g2ƒ Lp Rt R3x R3v ; for some 1 < p < 1, be such that ˇ

˛

.@t C v rx / D .1 t;x / 2 .1 v / 2 S ; for all 2 ƒ and for some bounded family fS .t; x; v/g2ƒ L1 Rt R3x R3v ; where ˛ 0 and 0 ˇ < 1. Then, for any 2 Cc1 R3 , the collection of velocity averages Z .t; x; v/ .v/ dv ; R3

is locally relatively compact in L1 Rt R3x .

2ƒ

128


Since each fk D M Gk is a weak solution of the renormalized equation (4.6), the preceding velocity averaging lemma implies, for any admissible nonlinearity ˇ.z/ 2 C 1 Œ0; 1/I R and any cutoff '.v/ 2 Cc1 R3 , treating F rv ˇ.Gk / D rv .Fˇ.Gk // as a source term, that Z ˇ.Gk /.t; x; v/'.v/ dv is relatively compact in L1loc .dtdx/: (4.12) R3

It is then possible to show that, up to further extraction of subsequences, Z Z fk v b.; /d ! f v b.; /d S2

as k ! 1 in

S2

L1loc .dtdxdv/

(4.13)

and almost everywhere,

where b.z; / may in fact be replaced by any collision kernel satisfying the DiPerna– Lions assumption (4.7). In particular, it follows that Q˙ .f; f / Q˙ .fk ; fk /

'.t; x/ !

'.t; x/ R R 1 C ıfk v S2 b.; /d 1 C ıf v S2 b.; /d as k ! 1 in w-L1 .dtdxdv/; for any '.t; x/ 2 Cc1 Œ0; 1/ R3 . At this point, using the convexity methods from [32], one can already establish the limiting entropy inequality (4.10), passing to the limit in (4.11). Lions further showed in [53], using Fourier integral operators, that the weak convergence of fk towards f in L1loc .dtdxdv/, the strong relative compactness of the velocity averages (4.12)–(4.13) and the uniform bounds from the entropy inequality (4.11) are sufficient to imply that, up to extraction of a subsequence, for every ı > 0, QC .f; f / QC .fk ; fk /

'.t; x/ !

'.t; x/ R R 1 C ıfk v S2 b.; /d 1 C ıf v S2 b.; /d as k ! 1 in L1 .dtdxdv/ and almost everywhere, for any '.t; x/ 2 Cc1 Œ0; 1/ R3 . Therefore, it holds in particular that QC .fk ; fk / ! QC .f; f / almost everywhere.

(4.14)

z Following [55], we fix now the specific renormalization ˇı .z/ D 1Cız , for any 0 < ı < 1, and we assume, without loss of generality, up to extraction of subsequences, that, as k ! 1,

Gk * ˇı ˇı .G/; 1 C ıGk 1 ˇı0 .Gk / D * hı ˇı0 .G/; .1 C ıGk /2 ˇı .Gk / D

ˇı0 .Gk /Gk D

Gk .1 C ıGk /2

D ˇı .Gk / .1 ıˇı .Gk // * gı ˇı .1 ıˇı / ;


129

in w -L1 loc .dtdxdv/. Therefore, passing to the limit in (4.6), we obtain, in view of the strong convergences (4.13) and (4.14), Z @t ˇı C v rx ˇı C F rv ˇı D hı QC .G; G/ gı f v b.; /d ; (4.15) S2

where the last term hı QC .G; G/ is well-defined in L1loc .dtdxdv/ by its mere nonnegativeness. Note that, for any > 0, choosing K > 0 large enough so that, by the equiintegrability of the Gk ’s, ; sup Gk 1fG Kg 1 Lloc .dt dxdv/

k

k2N

we find that kG ˇı kL1

loc .dt dxdv/

lim inf kGk ˇı .Gk /kL1

and

kG gı kL1

loc .dt dxdv/

loc .dt dxdv/

k!1

ıK sup kGk kL1 .dt dxdv/ C ; loc 1 C ıK k2N

lim inf Gk ˇı0 .Gk /Gk L1

loc .dt dxdv/

k!1 2 2

ı K C 2ıK .1 C ıK/2

sup kGk kL1

loc .dt dxdv/

k2N

C :

Hence, by the arbitrariness of > 0, lim kG ˇı kL1

D 0;

lim kG gı kL1

D 0:

ı!0

ı!0

loc .dt dxdv/ loc .dt dxdv/

Similarly, it is readily seen that, for any 1 p < 1, lim k1 hı kLp .dt dxdv/ D 0:

ı!0

loc

Finally, notice that ˇı , gı and hı are all increasing as ı vanishes. Hence, as ı ! 0, both ˇı and gı converge towards G almost everywhere, while hı converges towards a constant almost everywhere. Now comes a fundamental idea of Lions from [54, 55], which will be of particular interest to us and which has numerous qualitative consequences on renormalized solutions. This key idea consists in renormalizing equation (4.15) over again according to the following simple yet crucial lemma from [31]. Lemma 4.3 ([31, Theorem II.1, p. 516]). Let f .t; x/ 2 L1 .Œ0; T I Lploc .Rn //, with 1 < p 1, T > 0 and n 2 N, be a solution of the linear transport equation @t f C b rx f C cf D h;

(4.16)

130


where

1;˛ b 2 L Œ0; T I Wloc .Rn / ; c 2 L Œ0; T I L˛loc .Rn / ; h 2 L Œ0; T I Lˇloc .Rn / ;

for some p0 ˛ < 1, p1 C p10 D 1, 1 < 1 and 1 ˇ < 1 such that 1 D ˛1 C p1 . ˇ R Then, for any ı .x/ D ı1n xı , with 2 Cc1 .Rn /, 0, Rn .x/dx D 1 and ı > 0, the mollification fı D f ı satisfies @t fı C b rx fı C cfı D h C rı ; where the remainder rı vanishes in L .Œ0; T I Lˇloc .Rn //, as ı ! 0. In particular, it follows that, for any renormalization ˇ 2 C 1 .R/ such that ˇ 0 is bounded on R, @t ˇ.f / C b rx ˇ.f / C cf ˇ 0 .f / D hˇ 0 .f /: The above lemma has fundamental consequences in transport theory and in the theory of ordinary differential equations. Indeed, as established by DiPerna and Lions in [31], it can be shown that, loosely speaking, as soon as Lemma 4.3 applies, weak solutions of (4.16) are, in fact, renormalized solutions, unique and time continuous in the strong topology, and that the transport equation (4.16) propagates strong compactness. In turn, the properties of the transport equation have important consequences on ordinary differential equations, and the existence and uniqueness of a Lagrangian flow was also established in [31] under very weak assumptions on the 1;1 .Rn //. corresponding Eulerian flow, which should typically be in L1 .Œ0; T I Wloc Thus, in view of the regularity hypothesis (4.9) on the force field, 1;1 .dxdv/ ; F 2 L1loc dtI Wloc applying Lemma 4.3 to the transport equation (4.15) (transport by the vector field .v; F .t; x; v// 2 R6 ) yields that ˇı is a renormalized solutions of (4.15), that is to say, for any admissible renormalization ˇ, @t ˇ .ˇı / C v rx ˇ .ˇı / C F rv ˇ .ˇı / 0

C

0

D ˇ .ˇı / hı Q .G; G/ ˇ .ˇı / gı f v

Z

b.; /d :

(4.17)

S2

Finally, we let ı ! 0 in the above renormalized equation. To this end, notice that ˇ 0 .ˇı / gı is bounded uniformly by a constant pointwise and converges almost everywhere to ˇ 0 .G/G. Therefore, the last term above converges towards the expected


131

renormalized loss term while it remains uniformly locally integrable. Moreover, we see that, integrating (4.17) locally in all variables, the gain term ˇ 0 .ˇı / hı QC .G; G/ remains uniformly locally integrable, so that it converges towards the expected renormalized gain term. On the whole, since the left-hand side of (4.17) is easily handled by the strong convergence of ˇı towards G, we conclude, letting ı ! 0 in (4.17), that G solves (4.6) in the sense of distributions, which completes the justification of Theorem 4.1 according to [30, 32, 53, 54, 55]. It is to be emphasized that Theorem 4.1 can be easily generalized to a system of Vlasov–Boltzmann equations for two species of particles.

4.1.2 Coupling the Boltzmann equation with Maxwell’s equations Thus, we see that the validity of Theorem 4.1 rests crucially upon Lemma 4.3, and so 1;1 .dxdv// cannot be that the regularity hypothesis on the force field F 2 L1loc .dtI Wloc weakened, at least not with this method of proof. This is precisely the unique obstacle which prevents the construction of renormalized solutions for the Vlasov–Maxwell– Boltzmann systems (4.1) and (4.2), whose force fields are not regular. As far as the existence theory of global solutions is concerned, notice that the nonlinear coupling of a kinetic equation with Maxwell’s equations through the influence of a Lorentz force is not always a problem. In particular, it is possible to show the weak stability of the Vlasov–Maxwell system (without collisions) for densities in 2 L1 t Lx;v and, therefore, to establish the existence of (non-renormalized) weak solutions for this system (see [29]). Indeed, neglecting the collision operators in (4.1) and (4.2), the only remaining nonlinear terms are .E C v ^ B/ rv f

and

.E C v ^ B/ rv f ˙ :

Since the densities f in (4.1) and f ˙ in (4.2) do enjoy some strong compactness (even some kind of regularity) in time and space by virtue of velocity averaging lemmas (see [33], for instance), while the Lorentz force E C v ^ B is smooth in velocity (obviously, E and B do not depend on v), it is clear that the above nonlinear electromagnetic forcing terms are weakly stable as long as no renormalization is required. In conclusion, problematic difficulties arise when entering the realm of collisional kinetic theory, where renormalizing becomes a necessity. Nevertheless, it is to be noted that the existence of renormalized solutions for such collisionless Vlasov–Maxwell systems remains unknown, as well. In contrast with the Vlasov–Maxwell–Boltzmann systems, the Vlasov–Poisson– Boltzmann systems (2.35) and (2.85) do enjoy the existence of renormalized solutions (see [55]). Indeed, thanks to Poisson’s equation, the force fields therein have enough regularity to allow for an application of Lemma 4.3 and of the strategy of proof of Theorem 4.1.

132


Of course, since then, there have been generalizations of Lemma 4.3 and incidentally of the results from [31], most notably by Ambrosio [2], where the local Sobolev regularity of the vector field was relaxed to a local BV regularity, and by Le Bris and Lions [47], where a specific structure of the vector field, which unfortunately doesn’t 1;1 match the structure of (4.15), was used in order to impose a mere partial Wloc regularity on it. In any case, it is apparent, much like in the Cauchy–Lipschitz theorem on ordinary differential equations, that a minimum of a control on one full derivative of the vector field is necessary to crank the proof of Theorem 4.1, which is far from reach in the case of Vlasov–Maxwell–Boltzmann systems where E; B 2 L1 .dtI L2 .dx//, at best. This viewpoint is also corroborated by the counterexamples presented at the end of [31]. Thus, it seems that any result confirming the existence of renormalized solutions for Vlasov–Maxwell–Boltzmann systems will have to exploit the very specific structure of the electromagnetic interaction within the plasma. Surprisingly, the situation is much better when the microscopic interactions described by the collision operator have infinite range, so that the collisional crosssection has a singularity at grazing collisions: the entropy dissipation indeed controls some derivative with respect to v in this case. Using the hypoellipticity of the kinetic transport operator, we can then transfer part of this regularity onto the x variable. Following the strategy by Alexandre and Villani [1], and renormalizing the Vlasov– Boltzmann equation by concave functions, we thus get some global renormalized solutions involving a defect measure (which is formally 0 because of the conservation of mass). This construction has been sketched in [8]. It will be detailed and used to obtain fully rigorous convergence results in Part III. An alternative approach based on Young measures, as introduced by Lions in [55] will be the focus of our work in Part IV. We will see that, even though the notion of solution is very poor, the asymptotic analysis is robust and leads to similar convergence results. Note that Parts III and IV will be more technical as we will have to deal with very weak solutions. However the strategy of proof as well as the main arguments will be similar to the ones presented here, this is why we start with conditional results.

4.1.3 The setting of our conditional study We provide now a precise definition of renormalized solutions for the Vlasov– Maxwell–Boltzmann systems (4.1) and (4.2), even though their existence remains uncertain. Definition. We say that a density function f .t; x; v/ D M G.t; x; v/ 0 and electromagnetic vector fields E.t; x/ and B.t; x/, where .t; x; v/ 2 Œ0; 1/ R3 R3 , such that G 2 C Œ0; 1/I w-L1loc .dxdv/ \ L1 Œ0; 1/; dtI L1loc dxI L1 .1Cjvj2 /M dv ; E; B 2 C.Œ0; 1/I w-L2 .dx// \ L1 Œ0; 1/; dtI L2 .dx/ ;


133

are a renormalized solution of the one-species Vlasov–Maxwell–Boltzmann system (4.1) if they solve 8 @t ˇ .G/ C v rx ˇ .G/ C .E C v ^ B/ rv ˇ .G/ E vˇ 0 .G/ G ˆ ˆ ˆ ˆ D ˇ 0 .G/ Q.G; G/; ˆ ˆ Z ˆ ˆ ˆ ˆ ˆ @t E rot B D M Gv dv; < R3

ˆ @t B C rot E D 0; ˆ ˆ Z ˆ ˆ ˆ ˆ M G dv 1; div E D ˆ ˆ ˆ R3 ˆ : div B D 0; in the sense of distributions for any admissible renormalization, and satisfies the entropy inequality, for all t > 0, Z tZ Z 2 1 D.f /.s/ dxds jEj C jBj2 dx C H.f / C 2 R3 R3 0 (4.18) Z in 2 1 jE j C jB in j2 dx < 1; H f in C 2 R3 where f in D M G in is the initial value of f D M G and the relative entropy H.f / D H.f jM / is defined in (1.17), while the entropy dissipation D.f / is defined in (1.13). Definition. We say that density functions G C .t; x; v/ 0 and G .t; x; v/ 0, and electromagnetic vector fields E.t; x/ and B.t; x/, where .t; x; v/ 2 Œ0; 1/ R3 R3 , such that G ˙ 2 C Œ0; 1/I w-L1loc .dxdv/ \ L1 Œ0; 1/; dtI L1loc dxI L1 .1Cjvj2 /M dv ; E; B 2 C Œ0; 1/I w-L2 .dx/ \ L1 Œ0; 1/; dtI L2.dx/ ; are a renormalized solution of the two-species Vlasov–Maxwell–Boltzmann system (4.2) if they solve 8 ˆ @t ˇ G ˙ C v rx ˇ G ˙ ˙ .E C v ^ B/ rv ˇ G ˙ E vˇ 0 G ˙ G ˙ ˆ ˆ ˆ ˆ ˆ D ˇ 0 G ˙ Q.G ˙ ; G ˙ / C ˇ 0 G ˙ Q.G ˙ ; G /; ˆ ˆ Z ˆ ˆ ˆ ˆ M G C G v dv; < @t E rot B D R3

ˆ @t B C rot E D 0; ˆ ˆ Z ˆ ˆ ˆ ˆ ˆ M G C G dv; div E D ˆ ˆ ˆ R3 ˆ : div B D 0;

134


in the sense of distributions for any admissible renormalization, and satisfy for all t > 0 the entropy inequality Z 1 2 H fC CH f C jEj C jBj2 dx 2 R3 Z tZ C (4.19) D f C D .f / C D f C ; f .s/ dxds C 3 R 0 Z 1 in 2 jE j C jB in j2 dx < 1; H f Cin C H f in C 2 R3 where f ˙in D M G ˙in is the initial value of f ˙ D M G ˙ and the relative entropies H.f ˙ / D H.f ˙ jM / are defined in (1.17), while the entropy dissipations D.f ˙ / and D.f C ; f / are defined in (1.13) and (1.16).

4.1.4 Macroscopic conservation laws As already explained in Section 1.2, the one-species Vlasov–Maxwell–Boltzmann system (4.1) formally satisfies the macroscopic conservation laws 1 0 1 0 0 1 Z Z Z 1 v 0 @t f @ v A dv C rx f @v ˝ v A dv D f @E C v ^ B A dv; jvj2 jvj2 R3 R3 R3 Ev v 2

2

while the two-species Vlasov–Maxwell–Boltzmann system (4.2) formally satisfies the macroscopic conservation laws Z Z ˙ @t f dv C rx f ˙ v dv D 0; R3

R3

and Z C v˝v 2 dv dv C rx f Cf jvj v R3 R3 2 2 Z C E Cv^B D f f dv: Ev R3

Z @t

C f Cf

v

jvj2

However, it is at first unclear whether such formal laws are actually rigorously satisfied by the renormalized solutions defined in the previous section. It is therefore necessary to justify their validity. ˙To thisend, we suppose now that such renormalized solutions .f; E; B/ and f ; E; B have been previously obtained through an approximation procedure as detailed in Section 4.1.1. we assume that there are sequences ˚ More precisely, f.fk ; Ek ; Bk /gk2N and fk˙ ; Ek ; Bk k2N of smooth solutions to (4.1) and (4.2) (or appropriate approximations of these systems), for some uniformly bounded initial data, satisfying all macroscopic conservation laws and respectively converging in


135

some suitable weak sense towards .f; E; B/ and f ˙ ; E; B . Therefore, by virtue of the uniform bounds provided by the entropy inequalities (4.18) and (4.19), it is readily seen that the terms fk , fk v, fk˙ and fk˙ v are all respectively converging to f , f v, f ˙ and f ˙ v weakly in L1loc .dtdxI L1 .dv//. It follows that the conservations of mass Z Z @t f dv C rx f v dv D 0; R3 R3 Z Z @t f ˙ dv C rx f ˙ v dv D 0; R3

R3

are easily established for renormalized solutions. This is in general the case when dealing with collisional kinetic equations. However, the same is unfortunately not true for the conservations of momentum and energy. Indeed, these laws involve higher moments of f and f ˙ , which may be singular due to high velocities build-up, as well as products of electromagnetic fields with particle densities, which may not even make sense if not renormalized. To account for large velocities, we introduce now, following [57], since the terms fk jvj2 , f jvj2 , fk˙ jvj2 and f ˙ jvj2 are uniformly bounded in L1loc .dtdxI L1 .dv//, the Radon measures mij 2 Mloc .Œ0; 1/R3 /, i; j D 1; 2; 3, defined as the following defects in the limit k ! 1 (up to extraction of subsequences): Z Z fk vi vj dv * f vi vj dv C mij in Mloc Œ0; 1/ R3 ; R3

R3

in the one-species case, and Z Z C C fk C fk vi vj dv * f C f vi vj dvCmij R3

R3

in Mloc Œ0; 1/R3 ;

in the two-species case. Note that the measures mij are also defined by the limits, valid for any R > 0, Z Z fk vi vj 1fjvjRg dv * f vi vj 1fjvjRg dv C mij ; R3 R3 Z Z C C fk C fk vi vj 1fjvjRg dv * f C f vi vj 1fjvjRg dv C mij ; R3

R3

3 in loc .Œ0; 1/ R /. In particular, it follows that the matrix measure m D M mij 1i;j 3 is symmetric and positive definite in the sense that, for any ' 2 Cc Œ0; 1/ R3 I R3 ,

Z ' t .d m/' D Œ0;1/R3

3 Z X 3 i;j D1 Œ0;1/R

'i 'j d mij 0;

136


whence, for any '; 2 Cc Œ0; 1/ R3 I R3 , ˇ ˇZ Z Z ˇ 1 ˇ 1 t t ˇ ˇ ' .d m/ ˇ ' .d m/' C ˇ 2 Œ0;1/R3 2 Œ0;1/R3 Œ0;1/R3

t

.d m/ :

1

1

Further substituting ' and in the preceding inequality by 2 ' and 2 , with

> 0, respectively, and then optimizing the resulting inequality in > 0 yields that ˇZ ˇ ˇ ˇ

ˇ Z ˇ ' .d m/ ˇˇ 3

Œ0;1/R

t

Œ0;1/R3

t

12 Z

' .d m/'

t

12 .d m/

;

Œ0;1/R3

(4.20) for any '; 2 Cc Œ0; 1/ R3 I R3 . The matrix measure m will be used to characterize the flux terms in the conservation of momentum and the density terms in the conservation of energy. However, the flux terms in the conservation of energy contain higher order moments which cannot be handled and we will therefore simply leave them out of the analysis by only considering the global conservation of energy. As for the forcing terms involving the electromagnetic fields, they do not even make sense with the sole use of the a priori estimates provided by the entropy inequalities (4.18) and (4.19). It is therefore necessary to use now the conservation laws of energy (1.9)–(1.10) and for the Poynting vector (1.11)–(1.12) in Maxwell’s equations to recast these forcing terms with quadratic expressions involving the electromagnetic fields only. Thus, using the identities (1.9), (1.10), (1.11) and (1.12), the local conservations of momentum may be rewritten as Z Z @t fk v dv C Ek ^ Bk C rx fk v ˝ v dv Ek ˝ Ek Bk ˝ Bk R3 R3 jEk j2 C jBk j2 C rx D Ek ; 2 (4.21) in the one-species case, and as Z C @t fk C fk v dv C Ek ^ Bk R3 Z C C rx fk C fk v ˝ v dv Ek ˝ Ek Bk ˝ Bk (4.22) R3 jEk j2 C jBk j2 C rx D 0; 2 in the two-species case, whereas the global conservation of energy may be expessed as Z Z jEk j2 C jBk j2 jvj2 d fk dv C dx D 0; (4.23) dt R3 R3 2 2


in the one-species case, and as Z Z C 2 d jEk j2 C jBk j2 jvj dv C dx D 0; f C fk dt R3 R3 k 2 2

137

(4.24)

in the two-species case. Passing to the limit k ! 1 therefore requires the introduction of yet another set of Radon measures aij 2 Mloc .Œ0; 1/ R3 /, i; j D 1; 2; 3; 4; 5; 6, where the matrix measure a D .aij /1i;j 6 is defined as the following defect: Ek Ek E E ˝ * ˝ C a in Mloc Œ0; 1/ R3 : Bk Bk B B Note that the matrix measure a is also defined by the limit Ek E Ek E ˝ * a in Mloc Œ0; 1/ R3 : Bk B Bk B It then follows that, as before, the matrix measure a is symmetric and positive definite in the sense that, for any ' 2 Cc Œ0; 1/ R3 I R6 , Z ' t .da/' D Œ0;1/R3

6 Z X

'i 'j daij 0;

3 i;j D1 Œ0;1/R

whence (see the analogous inequality (4.20)) ˇZ ˇ ˇ ˇ

ˇ Z ˇ ' .da/ ˇˇ 3

Œ0;1/R

t

Œ0;1/R3

t

12 Z

' .da/'

t

12 .da/

; (4.25)

Œ0;1/R3

for any '; 2 Cc Œ0; 1/ R3 I R6 . For of notation, mere convenience we further introduce the matrix measures e D aij 1i;j 3 and b D a.i C3/.j C3/ 1i;j 3 . Now, letting k ! 1 in the conservation laws (4.21), (4.22), (4.23) and (4.24) respectively yields the local conservation of momentum 0 0 11 Z a26 a35 @t @ f v dv C E ^ B C @a34 a16 AA R3 a15 a24 Z f v ˝ v dv C m E ˝ E e B ˝ B b C rx R3 jEj2 C jBj2 C Tr a C rx D E; 2

138


in the one-species case, and 0 0 11 Z a26 a35 C f C f v dv C E ^ B C @a34 a16 AA @t @ R3 a15 a24 Z C f C f v ˝ v dv C m E ˝ E e B ˝ B b C rx R3 jEj2 C jBj2 C Tr a D 0; C rx 2 in the two-species case, as well as the global energy decay Z Z jvj2 Tr m jEj2 C jBj2 C Tr a f dv C C dx 2 2 2 R3 R3 Z Z 2 jE in j2 C jB in j2 in jvj f dv C dx; 2 2 R3 R3 in the one-species case, and Z Z C 2 Tr m jEj2 C jBj2 C Tr a jvj f Cf dv C C dx 2 2 2 R3 R3 Z Z Cin jvj2 jE in j2 C jB in j2 C f in f dv C dx; 2 2 R3 R3 in the two-species case. Note that the above global energy decay containing the defect measures may be incorporated into the entropy inequalities (4.18) and (4.19), so that renormalized solutions of the Vlasov–Maxwell–Boltzmann systems (4.1) and (4.2) may be assumed to respectively satisfy the entropy inequalities Z Z 2 1 1 2 Tr .m C e C b/ dx jEj C jBj dx C H.f / C 2 R3 2 R3 Z Z tZ 1 in 2 D.f /.s/ dxds H f in C jE j C jB in j2 dx < 1; C 2 R3 R3 0 (4.26) in the one-species case, and Z Z 1 2 1 H fC CH f C Tr .m C e C b/ dx jEj C jBj2 dx C 2 R3 2 R3 Z tZ C (4.27) D f C D .f / C D f C ; f .s/ dxds C R3 0 Z 1 in 2 jE j C jB in j2 dx < 1; H f Cin C H f in C 2 R3 in the two-species case.

139


The preceding characterization of defects in macroscopic conservation laws will not be of further use in our study of the hydrodynamic limits of the one-species Vlasov–Maxwell–Boltzmann system (4.1). It will, however, be of crucial utility in the renormalized relative entropy method developed later on in Chapter 12 in relation with hydrodynamic limits of the two-species Vlasov–Maxwell–Boltzmann system (4.2). We will then use a slightly more precise definition of renormalized solutions for the two-species Vlasov–Maxwell–Boltzmann system replacing (4.19) by (4.27). Notice, finally, that the symmetry and positive definiteness of the matrix measures m and a imply that the bounds on the non-negative measures Tr .m C e C b/, provided a priori by the entropy inequalities (4.26) and (4.27), are sufficient to control all components of m and a. Indeed, the inequalities (4.20) and (4.25) provide all necessary estimates of m and a in terms of Tr .m C e C b/. In particular, it is readily seen that, for any u 2 Cc R3 I R3 , ˇ 0 1 ˇ ˇ ˇZ Z 12 Z 12 a a 26 35 ˇ ˇ ˇ ˇ @ A .Tr e/ dx .Tr b/ dx : ˇ 3 u a34 a16 dx ˇ 6 kukL1 R3 R3 ˇ ˇ R a15 a24 Still, since the above defect stems from a vector product, it is possible to improve the constant in the preceding inequality. We record such improvement in the following result, for later use. Lemma 4.4. For any u 2 Cc R3 I R3 , it holds that ˇ 0 1 ˇ ˇ ˇZ Z 12 Z 12 a26 a35 ˇ ˇ ˇ ˇ @ A .Tr e/ dx .Tr b/ dx : ˇ 3 u a34 a16 dx ˇ kukL1 R3 R3 ˇ ˇ R a15 a24 Proof. Using (4.25), we first obtain ˇ 0 1 ˇ ˇZ ˇ a26 a35 ˇ ˇ ˇ @a34 a16 A dx ˇ u ˇ 3 ˇ ˇ R ˇ a15 a24 ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇˇ .a15 u3 a16 u2 / dx ˇˇ C ˇˇ .a26 u1 a24 u3 / dx ˇˇ 3 R3 ˇZ ˇ R ˇ ˇ C ˇˇ .a34 u2 a35 u1 / dx ˇˇ R3 Z 12 Z 12 2 2 a11 dx a55 u3 2a56 u3 u2 C a66 u2 dx R3

R3

12 Z

Z C R3

a22 dx

C R3

R3

12 Z

Z

a33 dx

R3

a66 u21 2a64 u1 u3 C a44 u23 dx a44 u22 2a45 u2 u1 C a55 u21 dx

12 12 :

140


It follows that, for any ˛ > 0, ˇ 0 1 ˇ ˇ ˇZ Z a26 a35 ˇ ˛ ˇ ˇ @a34 a16 A dx ˇ u ˇ 2 3 .Tr e/ dx ˇ 3 R ˇ ˇ R a15 a24 Z 1 a55 u23 2a56 u3 u2 C a66 u22 dx C 2˛ R3 Z 1 a66 u21 2a64 u1 u3 C a44 u23 dx C 2˛ R3 Z 1 a44 u22 2a45 u2 u1 C a55 u21 dx C 2˛ 3 Z R Z 1 ˛ .Tr e/ dx C .Tr b/ juj2 dx D 2 R3 2˛ R3 Z t 1 u bu dx 3 2˛ R Z Z kuk2L1 ˛ .Tr e/ dx C .Tr b/ dx; 2 R3 2˛ R3 which, upon optimizing in ˛ > 0, concludes the justification of the lemma.

4.2 The incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system Henceforth, in this second part of our work on conditional results, unless otherwise stated, we will focus, for the mere sake of technical simplicity, on some Maxwellian cross-section, say b 1. All other mathematically and physically pertinent crosssections (deriving from hard, soft, short-range and long-range interaction potentials) will be discussed and treated in full generality in the remaining parts of our work on unconditional results. Following Section 2.3, we first consider a plasma constituted of a gas of cations (positively charged ions), with a uniform background of heavy anions (negatively charged ions) assumed to be at statistical equilibrium. Elementary interactions are taken into account by both a mean field term (corresponding to long-range interactions) and a local collision term (associated to short-range interactions) involving possibly different mean free paths. Thus, the charged particles evolve under the coupled effect of the Lorentz force due to the self-induced electromagnetic field, and of the collisions with other particles, according to the following scaled Vlasov–Maxwell–


Boltzmann system: 8 1 ˆ ˆ @t f C v rx f C .E C v ^ B / rv f D Q.f ; f /; ˆ ˆ ˆ ˆ ˆ ˆ f D M .1 C g / ; ˆ ˆ ˆ Z ˆ ˆ ˆ < g vM dv; @t E rot B D ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

R3

141

(4.28)

@t B C rot E D 0; Z g M dv; div E D R3

div B D 0:

In this scaling, the entropy inequality states that Z Z Z 1 t 1 1 2 2 H f j C jB j D.f /.s/ dxds C jE dx C 2 2 R3 4 0 R3 Z in 2 1 1 jE j C jBin j2 dx; 2 H fin C 2 R3

(4.29)

where H.f / D H.f jM /. In particular, it yields uniform bounds on E , B and g . Since we are interested in the limiting fluctuation g * g, it is then natural to rewrite the kinetic equation in terms of the fluctuations g , 1 @t g C v rx g C .E C v ^ B / rv g E v .1 C g / D Lg C Q.g ; g /: (4.30) According to the formal analysis from Section 2.3, we then expect the limiting macroscopic observables to solve the incompressible quasi-static Navier–Stokes– Fourier–Maxwell–Poisson system: 8 @t u C u rx u x u D rx p C E C rx C u ^ B; ˆ ˆ ˆ ˆ ˆ div u D 0; ˆ ˆ ˆ < 5 3 3

x . C / D ; C u rx x D 0; @t ˆ 2 2 2 ˆ ˆ ˆ ˆ rot B D u; div E D ; ˆ ˆ ˆ : div B D 0: @t B C rot E D 0; (4.31) As discussed in Section 3.1, this system is similar to the usual Navier–Stokes equations, since it is weakly stable in the class of functions of finite energy.

142


Because of this crucial weak stability property, the study of hydrodynamic limits follows closely what has been previously done for the incompressible Navier–Stokes– Fourier limit of the Boltzmann equation (see [4, 70] and the references therein for a survey of related results). In particular, we will be able to prove a convergence result which: holds globally in time; does not require any assumption on the initial velocity profile; does not assume any constraint on the initial thermodynamic fields. We would also be able to take into account boundary conditions, and describe their limiting form, but this point will not be dealt with here. We refer to [61, 70] for a complete treatment of boundary conditions in the viscous hydrodynamic limits of the Boltzmann equation, based on the renormalized solutions on bounded domains constructed by Mischler in [62, 63]. As we will see, if we assume that the Vlasov–Maxwell–Boltzmann system (4.28) has renormalized solutions (which, again, is not known), the main challenge here lies in understanding the influence of the electromagnetic force both on hypoelliptic processes of the kinetic transport equation and on fast time-oscillations. Our goal, here, is to establish the convergence of scaled families of renormalized solutions to the one-species Vlasov–Maxwell–Boltzmann system (4.28) towards solutions of the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (4.31), without any restriction on their size, regularity or well-preparedness of the initial data. Following the program proposed by Bardos, Golse and Levermore in [11] (which relies essentially on weak compactness arguments), we can prove the following theorem. Recall that, in this second part, we are only considering the Maxwellian crosssection b 1. Theorem 4.5. Let fin ; Ein ; Bin be a family of initial data such that Z in 2 1 in 1 in 2 H f j C jB j (4.32) C jE dx C in ; 2 2 R3 for some C in > 0, and Z div Ein

D R3

gin M dv;

div Bin D 0;

(4.33)

where fin D M 1 C gin . For any > 0, we assume the existence of a renormalized solution .f ; E ; B / to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) (for the Maxwellian cross-section b 1) with initial data fin ; Ein ; Bin . We define the macroscopic fluc-


143

tuations of density , bulk velocity u and temperature by Z D g M dv; R3 Z u D g vM dv; R3 2 Z jvj g D 1 M dv; 3 R3 in and denote their respective initial value by in , uin and . Then, the family . ; u ; ; B / is weakly relatively compact in L1loc .dtdx/ (while in in 1 the family of initial data in ; uin ; ; B is weakly relatively compact in Lloc .dx/) and any of its limit points .; u; ; B/ is a weak solution of the incompressible quasistatic Navier–Stokes–Fourier–Maxwell–Poisson system (4.31) with initial data

x in rot in 30 20in ; uin D rot uin 0 C B0 ; 3 5 x 1 x 1

1 x in 30in 20in ; B in D rot uin in D 0 C B0 ; 3 5 x 1 x in in in in in in 2 where 0in ; uin 0 ; 0 ; B0 2 L .dx/ is the weak limit of ; u ; ; B . in D

The proof of Theorem 4.5 is built over the course of the coming chapters and is per se the subject of Chapter 11. Note that, strictly speaking, the weak solution we obtain in the limit (and which depends in general on the subsequence under consideration) is not necessarily a Leray solution of the system (4.31), since it is does not satisfy the energy inequality (3.2), but only a bound. However, it is possible to obtain asymptotically a Leray solution by strengthening the initial well-preparedness of the data. More precisely, one would have to impose that the initial data converge entropically (as introduced in [11]) in the sense that Z in 2 1 in 1 H f C jE j C jBin j2 dx 2 2 3 Z R in 2 ˇ in ˇ2 3 in 2 ˇ in ˇ 1 in ˇ2 in 2 ˇ ˇ ˇ C u C C rx C C jB j dx: ! 2 R3 2 In fact, the above entropic convergence has rather strong implications on the initial data. Indeed, further denoting by g0in and E0in the weak limits of gin and Ein , respectively, standard convexity arguments on weak convergence (see Lemma 5.1 below, or

144


[11, Proposition 3.1]) yield that Z Z ˇ ˇ2 3 in 2 in 2 1 1 2 ˇ C 0in C ˇuin dx D …g0 M dvdx 0 0 2 R3 2 2 R3 R3 Z in 2 1 1 g0 M dvdx lim inf 2 H fin ; !0 2 R3 R3 Z Z ˇ in ˇ2 ˇ in ˇ2 ˇ in ˇ2 ˇ in ˇ2 1 ˇE ˇ C ˇB ˇ dx lim inf 1 ˇE ˇ C ˇB ˇ dx: 0 0 !0 2 R3 2 R3 Moreover, one easily verifies that in in 0 1 0 in 1

x 3 div E0in C 35

3 20in 35

xq 0 q0 qx 0 in C B 2 B 3 in C 1 x 3 3 @ 2 0 A 7! @ 35

in div E0in C 35

30 20in A 2 0 2 x x 5 1 E0in r 0in div E0in C 35

rx 30in 20in 35 x x x and

defines the orthogonal projection onto the subspace of L2 .dx/ defined by the constraint rx 0in C 0in D E0in ; while

uin 0 B0in

7!

rot 1 x 1 1 x

! in C PB rot uin 0 0 in rot uin 0 C PB0

corresponds to the orthogonal projection onto the subspace of L2 .dx/ defined by the constraints rot B0in D uin and div B0in D 0: 0 Therefore, it follows that Z in 2 ˇ in ˇ2 3 in 2 ˇ in ˇ2 ˇ ˇ2 1 C ˇu ˇ C C ˇrx C in ˇ C ˇB in ˇ dx 2 R3 2 Z ˇ ˇ in 2 1 3 in 2 ˇˇ in ˇˇ2 ˇˇ in ˇˇ2 in ˇ2 ˇ 0 C u0 C dx; C E0 C B0 2 R3 2 0 in in in in D E0in and with equality if and only if in D 0in , uin D uin 0 , D 0 , rx C in in the initial B D B0 , which, when combined with the above entropic convergence in in of in in in data and according to Proposition 4.11from [11], implies that ; u ; ; E ; B converges strongly to in ; uin ; in ; rx in C in ; B in , where div uin D 0; div B in D 0; rot B in D uin ;

x in C in D in ; (4.34) and that gin converges strongly to 2 3 jvj in C uin v C in ; in L1loc dxI L1 1 C jvj2 Mdv : 2 2


145

The strong convergence of gin towards an infinitesimal Maxwellian implies the vanishing of the initial relaxation layer, while the strong convergence of the initial macroscopic observables towards initial data satisfying the constraints (4.34) implies that there are no acoustic-electromagnetic waves. In fact, the weak convergence result in Theorem 4.5 could be strengthened into a strong convergence result, for wellprepared initial data and provided that the limiting system has a unique solution satisfying the energy equality (see [11, Theorem 7.4] on the strong Navier–Stokes limit). It is to be emphasized that the generalized relative entropy method, which is developed later on in Chapter 12 and is used to prove Theorems 4.6 and 4.7 below, is also applicable to the asymptotic regime studied in Theorem 4.5. This method would provide some strong convergence result even for ill-prepared initial data provided we can build an approximate solution which is smooth and accounts for the corrections due to the initial layer and the acoustic-electromagnetic waves.

4.3 The two-fluid incompressible Navier–Stokes–Fourier– Maxwell system with (solenoidal) Ohm’s law According to Section 2.4, we consider now a plasma constituted of two species of oppositely charged particles with approximately equal masses, namely cations (positively charged ions) and anions (negatively charged ions). Elementary interactions are taken into account by both mean field terms (corresponding to long-range interactions) and some local collision terms (associated to short-range interactions) involving possibly different mean free paths. Thus, the charged particles evolve under the coupled effect of the Lorentz force due to the self-induced electromagnetic field, and of the collisions with other particles, according to the following scaled two-species Vlasov–Maxwell–Boltzmann system: 8 1 ˙ ˙ ˙ ˙ ˙ ˆ ˆ ˆ @t f C v rx f ˙ ı .E C v ^ B / rv f D Q.f ; f / ˆ ˆ ˆ ˆ ˆ ı2 ˆ ˆ Q.f˙ ; f /; C ˆ ˆ ˆ ˆ ˆ ˆ ˆ f˙ D M 1 C g˙ ; ˆ < Z C ı @t E rot B D g g vM dv; ˆ ˆ 3 ˆ R ˆ ˆ ˆ ˆ B C rot E D 0; @ ˆ t ˆ Z ˆ ˆ C ˆ ˆ ˆ D ı g g M dv; div E ˆ ˆ ˆ R3 ˆ : div B D 0; (4.35)

146 where that

4 Two typical regimes ı

is asymptotically unbounded. In this scaling, the entropy inequality states

Z Z 1 1 1 1 C H f C 2 H f C 2 Tr m dx C jE j2 C jB j2 C Tr a dx 2 2 R3 2 R3 Z Z C 1 t C 4 D f C D f C ı2 D fC ; f .s/ dxds 0 R3 Z 1 in 2 1 1 Cin C 2 H fin C jE j C jBin j2 dx; 2 H f 2 R3 (4.36) m where H.f˙ / D H.f˙ jM / and the symmetric positive definite matrix measures R and a are the defects introduced in Section 4.1.4 stemming from the terms R3 .fC C f /v ˝ v dv and E E ˝ ; B B respectively. In particular, it yields uniform bounds on E , B and g˙ . Since we are interested in the limiting fluctuation g˙ * g ˙ , it is then natural to rewrite the kinetic equations in terms of the fluctuations g˙ , C C C g g g 1 C gC C v rx C ı.E C v ^ B / rv ıE v @t g g g 1 g C C C 2 C 1 Lg C ı2 L gC ; g ; g : C C Q.g ; g / C ı2 Q g D 2 C Q.g ; g / C ı Q g ; g L g C ı L g ; g (4.37) According to the formal analysis from Section 2.4, we then expect the limiting macroscopic observables to solve, in the case of strong interspecies collisions ı D 1, the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law: 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C .nE C j ^ B/ ; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; div E D n; @t E rot B D j; (4.38) ˆ ˆ ˆ @t B C rot E D 0; div B D 0; ˆ ˆ ˆ ˆ 1 ˆ ˆ : w D n; j nu D rx n C E C u ^ B ; 2 and, in the case of weak interspecies collisions ı D o.1/, with ı unbounded, the twofluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s


law: 8 1 ˆ ˆ div u D 0; @t u C u rx u x u D rx p C j ^ B; ˆ ˆ 2 ˆ ˆ ˆ ˆ C D 0; ˆ < @t C u rx x D 0; div E D 0; @t E rot B D j; ˆ ˆ @ B C rot E D 0; div B D 0; ˆ t ˆ ˆ ˆ ˆ j D .rx pN C E C u ^ B/ ; div j D 0; ˆ ˆ : n D 0; w D 0:

147

(4.39)

As previously emphasized in Section 3.2, the above limiting models (4.38) and (4.39) are not stable under weak convergence in the energy space and, thus, share more similarities with the three-dimensional incompressible Euler equations. Our goal here is to establish the convergence of scaled families of renormalized solutions to the two-species Vlasov–Maxwell–Boltzmann system (4.35) towards dissipative solutions of the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law (4.38), in the case of strong interspecies interactions ı D 1, or with solenoidal Ohm’s law (4.39), in the case of weak interspecies interactions ı D o.1/, with ı unbounded, without any restriction on their size or regularity. We will, however, impose some well-preparedness of the initial data. Improving on the program by the second author completed in [69, 71] (which relies essentially on modulated entropy arguments), we can prove the following theorems. Recall that, in this second part, we are only considering the Maxwellian crosssection b 1.

4.3.1 Weak interactions We first define precisely the kind of dissipative solution of (4.39) that is considered here. Definition. We say that .; u; ; n; j; w; E; B/ 2 L1loc .dtdx/; is a dissipative solution of the two-fluid incompressible Navier–Stokes–Fourier– Maxwell system with solenoidal Ohm’s law (4.39) if: it verifies the energy inequality corresponding to (4.39), it enjoys the weak temporal continuity .u; ; E; B/ 2 C Œ0; 1/I w-L2 R3 ;

148


it solves the system 8 div u D 0; C ˆ ˆ ˆ ˆ div E ˆ < @t E rot B D j; @t B C rot E D 0; div B ˆ ˆ ˆ j D .rx pN C E C u ^ B/ ; div j ˆ ˆ : n D 0; w

D 0; D 0; D 0; D 0; D 0;

in the sense of distributions, it satisfies the stability inequality Z Rt 1 t ıE .t/ C ıD .s/e s . /d ds 2 0 0 ıE .0/e

Rt 0

.s/ds

1 u uN 5 C B Z tZ N Rt 2 C B C .s/ dxe s . /d ds; B N j j C AB C R3 0 @E EN C uN ^ B BN A B BN C E EN ^ uN

N jN; E; N B/ N 2 Cc1 .Œ0; 1/ R3 / with for any test functions .u; N ; div uN D div jN D div EN D div BN D 0

and

N L1 .dt dx/ < 1; kuk

where the modulated energy and modulated energy dissipation are respectively given by 5 1 N 22 N 2L2 .dx/ C k N k2L2 .dx/ C kE Ek ıE .t/ D ku uk L .dx/ 2 Z 2 1 N ^ .B B// N uN dx; N 22 C kB Bk ..E E/ L .dx/ 2 R3 N 2 2 C 1 kj jNk2 2 ; N 2L2 C 5krx . /k ıD .t/ D 2krx .u u/k Lx Lx x the acceleration operator is defined by 1 0 2 .@t uN C P.uN rx u/ N x u/ N C P jN ^ BN B C 2 @t N C uN rx N xN C B 1 N N B C; N N N N N A u; N ; j ; E; B D B j C P E C uN ^ B C @ A @tEN rot BN CjN @t BN C rot EN


149

and the growth rate is given by

.t/ D C

N N N ku.t/k W 1;1 .dx/ C k@t u.t/k L1 .dx/ C j .t/ L1 .dx/

C N .t/W 1;1 .dx/

1 ku.t/k N L1 .dx/ ! 2 N ; C .t/ 1;1 W

.dx/

with a constant C > 0 independent of test functions. Remark. This dissipative solution coincides with the unique smooth solution of (4.39) with velocity field bounded pointwise by the speed of light (i.e., kukL1 .dt dx/ < 1) as long as the latter exists. As previously mentioned, the result presented here builds upon the work by the second author [69, 71], which requires in particular that we impose some wellpreparedness of the initial data. We introduce now carefully the requirements on the initial data that are to be considered in the theorem below concerning weak interspecies interactions. Definition. We say that a family of initial data f˙in ; Ein ; Bin such that Z in 2 1 in 1 1 Cin in 2 H f H f j C jB j C C jE dx C in ; 2 2 2 R3 for some C in > 0, and div Ein D ı

Z R3

Cin g gin M dv;

div Bin D 0;

(4.40)

where f˙in D M 1 C g˙in , is well-prepared (for the setting of weak interspecies interactions) if 2 3 ˙in in in in in jvj g * g D C u v C in w-L1loc .Mdxdv/; 2 2 as ! 0, where in ; uin ; in 2 L2 .dx/ satisfy the incompressibility and Boussinesq constraints div uin D 0; in C in D 0; and if the following entropic convergences hold Z in 2 1 Cin 1 in H f H f M dvdx; C ! g 2 2 R3 R3 Ein ! E in ; in L2 .dx/; Bin ! B in ; as ! 0, for some E in ; B in 2 L2 .dx/.

in L2 .dx/;

(4.41)

150


Remark. In the definition above, observe that it necessarily holds that div E in D 0;

div B in D 0;

as a consequence of (4.40). In the case of weak interspecies interactions ı D o.1/, with ı unbounded, we have the following result. Theorem 4.6. Let f˙in ; Ein ; Bin be a family of initial data such that Z in 2 1 in 1 1 Cin in 2 H f H f j C jB j (4.42) C C jE dx C in ; 2 2 2 R3 for some C in > 0, and div Ein

Z

Dı R3

gCin gin M dv;

div Bin D 0;

where f˙in D M 1 C g˙in . We further assume that the initial data are wellprepared in the sense of the definition above. For any > 0, we assume the existence of a renormalized solution f˙ ; E ; B to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) (for the Maxwellian cross-section b 1), where ı D o.1/ and ı is asymptotically un bounded, with initial data f˙in ; Ein ; Bin . We define the macroscopic fluctuations ˙ of density ˙ , bulk velocity u˙ and temperature by Z ˙ g˙ M dv; D 3 ZR ˙ u D g˙ vM dv; 3 R 2 Z ˙ ˙ jvj g D 1 M dv: 3 R3 We finally define the hydrodynamic variables C C ; 2 and electrodynamic variables D

u D

uC C u ; 2

D

C C ; 2

ı C ı C w D u u : ; Then, the family . ; u ; ; n ; j ; w ; E ; B / is weakly relatively compact in L1loc .dtdx/ and any of its limit points .; u; ; n; j; w; E; B/ is a dissipative solution of the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law (4.39) with initial data uin ; in ; E in ; B in . n D C ;

j D

The proof of Theorem 4.6 is built over the course of the coming chapters and is per se the subject of Section 12.2.


151

4.3.2 Strong interactions We first define precisely the kind of dissipative solution of (4.38) that is considered here. Definition. We say that .; u; ; n; j; w; E; B/ 2 L1loc .dtdx/; is a dissipative solution of the two-fluid incompressible Navier–Stokes–Fourier– Maxwell system with Ohm’s law (4.38) if: it verifies the energy inequality corresponding to (4.38), it enjoys the weak temporal continuity .u; ; n; E; B/ 2 C Œ0; 1/I w-L2 R3 ; it solves the system 8 div u D 0; ˆ ˆ ˆ ˆ ˆ div E D n; < ˆ @t B C rot E D 0; ˆ ˆ ˆ ˆ :

C D 0; div B D 0;

1 j nu D rx n C E C u ^ B ; 2 w D n;

in the sense of distributions (note that we have voluntarily left Ampère’s equation out of the above system), it satisfies the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 ıE .0/e

Rt

0

.s/ds

1 u uN 5 C B Z tZ N Rt 2 C B C .s/ dxe s . /d ds; N nN uN j nu j C AB C B R3 0 @E EN C uN ^ B BN 1 rx .n n/ N A 2 N N B B C E E ^ uN 0

N n; N BN 2 Cc1 .Œ0; 1/ R3 / with for any test functions u; N ; N jN ; E; div uN D div BN D 0;

div EN D nN

and

N L1 .dt dx/ < 1; kuk

152


where the modulated energy and modulated energy dissipation are respectively given by 1 5 N 2L2 .dx/ C k N k2L2 .dx/ kn nk N 2L2 .dx/ C ku uk 4 2 1 1 N 22 N 22 C kE Ek C kB Bk .dx/ L L .dx/ 2 2 Z E EN ^ B BN uN dx; R3 ıD .t/ D 2krx .u u/ N k2L2 C 5krx N k2L2 ıE .t/ D

x

1 C k .j nu/ jN nN uN k2L2 ; x

x

the acceleration operator is defined by 1 0 2 .@t uN C P .uN rx u/ N x u/ N C P nN EN C jN ^ BN B N N N C C B 2 @t C uN 1 rx x 1 N N B C; N N N N N A u; N ; j ; E; B D B j nNuN 2 rx nN C E C uN ^ B C @ A @tEN rot BN CjN @t BN C rot EN and the growth rate is given by

.t/ D C

N N N u/.t/k N N ku.t/k N L1 .dx/ C k .t/kW 1;1 .dx/ C k.j n L1 .dx/ W 1;1 .dx/ C k@t u.t/k 1 ku.t/k N L1 .dx/ ! 1 N N C kN .t/k2W 1;1 .dx/ C ; 2 rx nN E uN ^ B .t/ 1 L .dx/ with a constant C > 0 independent of test functions.

Remark. This dissipative solution coincides with the unique smooth solution of (4.38) with velocity field bounded pointwise by the speed of light (i.e., kukL1 .dt dx/ < 1) as long as the latter exists. As before, we need to impose some well-preparedness of the initial data. We introduce now carefully the requirements on the initial data that are to be considered in the theorem below concerning strong interspecies interactions. Definition. We say that a family of initial data f˙in ; Ein ; Bin such that Z in 2 1 in 1 1 Cin in 2 H f H f j C jB j C C jE dx C in ; 2 2 2 R3

153


for some C in > 0, and div Ein

Z D R3

Cin g gin M dv;

div Bin D 0;

(4.43)

where f˙in D M 1 C g˙in , is well-prepared (for the setting of strong interspecies interactions) if 2 3 ˙in ˙in ˙in in in jvj g * g D Cu vC in w-L1loc .Mdxdv/; 2 2 as ! 0, where ˙in ; uin ; in 2 L2 .dx/ satisfy the incompressibility and Boussinesq Cin in constraints, denoting in D C , 2 div uin D 0;

in C in D 0;

and if the following entropic convergences hold Z 1 in 1 Cin 2 1 in 2 1 Cin H f C M dvdx; C 2 H f ! g g 2 2 2 R3 R3 Ein ! E in ;

in L2 .dx/;

Bin ! B in ;

in L2 .dx/; (4.44)

as ! 0, for some E in ; B in 2 L2 .dx/.

Remark. In the definition above, observe that it necessarily holds, denoting nin D Cin in , that div E in D nin ; div B in D 0; as a consequence of (4.43). In the case of strong interspecies interactions ı D 1, we have the following result. Theorem 4.7. Let f˙in ; Ein ; Bin be a family of initial data such that Z in 2 1 in 1 1 Cin in 2 H f H f j C jB j (4.45) C C jE dx C in ; 2 2 2 R3 for some C in > 0, and Z div Ein D

R3

Cin g gin M dv;

div Bin D 0;

where f˙in D M.1 C g˙in /. We further assume that the initial data are wellprepared in the sense of the definition above. For any > 0, we assume the existence of a renormalized solution f˙ ; E ; B to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) (for the

154


Maxwellian cross-section b 1), where ı D 1, with initial data f˙in ; Ein ; Bin . We define the macroscopic fluctuations of density ˙ , bulk velocity u˙ and temperature ˙ by Z ˙ D

u˙ D

R3

Z

g˙ M dv;

g˙ vM dv; 2 Z ˙ jvj D g 1 M dv: 3 R3

˙

R3

We finally define the hydrodynamic variables D

C C ; 2

uC C u ; 2

u D

D

C C ; 2

and electrodynamic variables n D C ;

j D

1 C u u ;

w D

1 C :

Then, the family . ; u ; ; n ; j ; E ; B / (note that we have excluded the variable w ) is relatively compact in the sense that for every sequence in this family there exists a subsequence such that . ; u ; ; n ; r j ; E ; B / * .; u; ; n; j; E; B/

in w-L1loc .dtdx/;

where r .t; x/ is a sequence of mesurable scalar functions converging almost everywhere towards the constant function 1. Moreover, up to further extraction of subsequences, one also has the convergence r We set

1 C g g n * h

in w-L1loc dtdxI L1 ..1 C jvj/Mdv/ :

jvj2 h 1 M dv: wD 3 R3 Z

Note that w is not necessarily a limit point of w . Any such limit point .; u; ; n; j; w; E; B/ is a dissipative solution of the twofluid incompressible system with Ohm’s law (4.38) Navier–Stokes–Fourier–Maxwell with initial data uin ; in ; nin ; E in ; B in . The proof of Theorem 4.7 is built over the course of the coming chapters and is per se the subject of Section 12.3. In both Theorems 4.6 and 4.7, we focus on the case of well-prepared initial data. That is to say, we assume that the initial distribution has a velocity profile close to

4.4 Outline of proofs

155

local thermodynamic equilibrium g ˙in D ˙in C uin v C in

jvj2 3 ; 2 2

(with g Cin D g in in the case of weak interactions) in order that there is no relaxation layer, and that the asymptotic initial thermodynamic fields satisfy the incompressibility and Boussinesq constraints div uin D 0;

in C in D 0;

which ensures that there are no acoustic waves. The case of ill-prepared initial data could be handled by constructing an accurate approximate solution as in [71]. The corresponding result should be even better in the present viscous incompressible regime because we can control conservation defects and fluxes without any additional integrability assumptions on renormalized solutions to (4.35). Note, however, that such a result would still be conditional, as the existence of renormalized solutions to (4.35) has to be assumed. Relaxing the regularity assumption on the asymptotic solution would require new ideas: the stability in the energy and entropy methods is indeed controlled by higher integrability or regularity norms of the limiting fields. As discussed in Section 3.2, the two-fluid incompressible Navier–Stokes–Fourier–Maxwell systems with (solenoidal) Ohm’s law (4.38) and (4.39) are not known to have weak solutions, so that we do not expect to extend our convergence results for distributional solutions with low regularity.

4.4 Outline of proofs We expect the Vlasov–Maxwell–Boltzmann systems (4.28) and (4.35) to exhibit very different qualitative behaviors in the three asymptotic scalings we consider: one species, two species with weak interactions, and two species with strong interactions. However, estimates coming directly from the entropy inequalities (4.29) and (4.36) and leading to weak compactness results are similar in all regimes, so we will gather them in Chapter 5. We will also obtain the thermodynamic equilibria coming from relaxation estimates in Chapter 5. Then, Chapter 6 will be devoted to the derivation of constraints which are stable under weak convergence and can be handled with the weak bounds from Chapter 5. These constraints include, for instance, some lower order macroscopic constraints (such as the Boussinesq and incompressibility relations) for one species or two species with weak interactions. We will also establish the limiting energy inequalities for one species and two species with weak interactions and discuss the limiting form of Maxwell’s system. This chapter does not handle the constraints pertaining to two species with strong interactions, which will require the more advanced techniques of the following chapters.

156


A major difference between regimes appears in Chapter 7 regarding spatial regularity. The basic idea is to use the hypoellipticity of the free transport operator – as studied in [7] – to transfer regularity from the v variable to the x variable. But, because of the singularity in the Lorentz force, source terms in the kinetic equations are of different sizes, so that different renormalizations of the kinetic equations will have to be considered for the three different regimes. Roughly speaking, we will be able to establish some strong compactness and equi-integrability on the fluctuations in the less singular regime with only one species, and only some weaker analog on some truncated fluctuations for two species (with a truncation depending on the asymptotic parameter ı). Another important difference comes from the nonlinear constraints (which occur only in the cases of two species). We will see in Chapter 8 that, for weak interactions, Ohm’s law is obtained as a higher-order singular perturbation. Its derivation will use the renormalized form of the kinetic equations as well as the partial equi-integrability established in Chapter 7. In the case of strong interactions, even though Ohm’s law appears at leading order, proving its stability is much more intricate as it will require an additional macroscopic renormalization. Chapter 8 will also contain the derivation of the energy inequality for strong interactions, which will require the use of the strong compactness bounds from Chapter 7, as well. The last pieces of information we will need to get the consistency of the hydrodynamic limits are the approximate conservation laws, for which we have to go even further in the asymptotic expansions (see Chapter 9). In the case of one species, this will require to use some suitable renormalization as well as the equi-integrability established in Chapter 7. In the case of two species, such strong equi-integrability properties are no longer available and we will have to rely on weaker bounds. Thus, in this case, we will only obtain a conditional result, in the sense that the conservation defects and remainders will be controlled by some modulated entropy. We will therefore need, later on (in Chapter 12), some loop argument based on Grönwall’s lemma to prove both the consistency and the convergence in these regimes. In view of these differences, the convergence proofs will follow different strategies. In the case of one species, we will use a weak compactness method which relies on some precise study of acoustic and electromagnetic waves (described in Chapter 10) and compensated compactness. The core of the proof of convergence for one species will then be the content of Chapter 11. For two species with both weak and strong interactions, we will finally introduce in Chapter 12 a novel renormalized relative entropy method, which will allow to get some stability without any a priori spatial regularity.

Chapter 5

Weak compactness and relaxation estimates In this chapter, we establish and recall, from previous works on the hydrodynamic limit of the Boltzmann equation, essential weak compactness estimates on the fluctuations, based on the uniform bounds provided by the scaled relative entropy inequalities (4.29), in the case of Theorem 4.5, and (4.36), in the case of Theorems 4.6 and 4.7. The results presented here are somewhat preliminary to the core of the proofs of Theorems 4.5, 4.6 and 4.7. Thus, they include the first rigorous steps in the proofs of our main theorems and are sometimes straightforward adaptations of lemmas from previous works on the hydrodynamic limit of the Boltzmann equation, while some are new or non-trivial adaptations. In particular, the estimates for two species of particles presented below are all novel. First, recall that we are considering, in Theorem 4.5, a sequence of renormalized solutions .f ; E ; B / of the Vlasov–Maxwell–Boltzmann scaled one-species system (4.28) with initial data fin ; Ein ; Bin satisfying the uniform bound (4.32), in Theorems 4.6 and 4.7 we consider a sequence of renormalized solutions while f˙ ; E ; B of the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) with initial data f˙in ; Ein ; Bin satisfying the uniform bound (4.42). We will conveniently employ the notations for fluctuations f D M G D M .1 C g / ; fin D M Gin D M 1 C gin ; f˙ D M G˙ D M 1 C g˙ ; f˙in D M G˙in D M 1 C g˙in ; and for scaled collision integrands q D qC D q D qC; D q;C D

1 2 1 2 1 2 ı 2 ı 2

0 0 G G G G ;

0 C0 C GC G GC G ; 0 0 G G G G ;

0 0 GC G ; GC G

C0 C G G G 0 G :

158

5 Weak compactness and relaxation estimates

5.1 Controls from the relative entropy bound Following [11], we introduce the non-negative convex function h.z/ D .1 C z/ log.1 C z/ z; defined over .1; 1/. We may then recast the entropy inequalities (4.29) and (4.36) utilizing this notation to get the relative entropy bounds, for all t 0, Z 1 1 1 H f H f jM D h .g / M dxdv C in ; (5.1) D 2 3 3 2 2 R R and 1 1 ˙ H f D 2 H f˙ jM D 2

Z R3 R3

1 ˙ h g M dxdv C in : 2

(5.2)

The relative entropy bounds are expected to control the size of the fluctuations g and g˙ since 1 h.z/ z 2 ; near z D 0: 2 However, this behavior only holds asymptotically, as z ! 0, and thus, in order to exploit the relative entropy bounds, we will have to rely crucially on Young’s inequality (B.3) for h.z/, presented in Appendix B. The following lemma is a mere reformulation of Proposition 3.1 from [11]. It is a consequence solely of the fact that the fluctuations satisfy the entropy bounds (5.1) and (5.2). Lemma 5.1. Let f .t; x; v/ be a family of measurable, almost everywhere nonnegative distribution functions such that, for all t 0, Z 1 1 H f h .g / .t/M dxdv C in : .t/ D 2 3 3 2 R R Then, as ! 0: (1) any subsequence of fluctuations g is uniformly bounded in L1 dtI L1loc dxI L1 1 C jvj2 M dv ; and weakly relatively compact in L1loc dtdxI L1 1 C jvj2 M dv : (2) if g is a weak limit point in L1loc dtdxI L1 1 C jvj2 Mdv of the family of fluctuations g , then g belongs to L1 .dtI L2 .M dxdv// and satisfies, for almost every t 0, Z 1 g 2 .t/M dxdv C in : 2 R3 R3

5.1 Controls from the relative entropy bound

159

Proof. For the sake of completeness, we recall the main ideas from the proof of Proposition 3.1 in [11], which is based on an application of inequality (B.3). Thus, setting y D 14 1 C jvj2 , z D g , ˇ D 4 and ˛ 4 in (B.3) yields, almost everywhere in .t; x; v/, 1

˛ 16e 4 jvj2 e 4 : 1 C jvj2 jg j 2 h .g / C ˛ This is then integrated in all variables on suitable sets to demonstrate, with the entropy bound, the equi-integrability and tightness of the sequences, and thus their weak compactness. We set ˛ D 1 first. Then, for each measurable set E R3 of finite measure, it holds that, for every 0 < 14 , i.e., for all but a finite number of ’s, Z Z jvj2 1 .1 C jvj2 / jg .t/j M dxdv C in C 16e 4 jEj e 4 M dv: E R3

R3

Hence, the family 1Cjvj2 g is uniformly bounded in L1 dtI L1loc dxI L1 .M dv/ . Similarly, for arbitrary ˛ 4 and for any measurable set E Œ0; T R3 R3 , where T > 0, one has that Z

1

16e 4 .1 C jvj / jg j M dtdxdv ˛T C C jEj: ˛ E 2

in

This shows, by the arbitrariness of ˛ 4, that the family .1Cjvj2 /g M is uniformly integrable on Œ0; T R3 R3 . Finally, for arbitrary ˛ 4, any time T > 0, any compact set K R3 and any large radius R > 0, we find that Z

1

16e 4 T jKj 1 C jvj2 jg j M dtdxdv ˛T C in C ˛ Œ0;T KfjvjRg

Z e

jvj2 4

M dv;

fjvjRg

which, by the arbitrariness of ˛ 4, clearly implies the tightness in velocity of the family .1 C jvj2 /g M . On the whole, by virtue of the Dunford–Pettis criterion [68], we infer the weak relative compactness of the family .1 C jvj2 /g M in L1loc .dtdxI L1 .dv//, which concludes the demonstration of the first assertion of the lemma. The second assertion will follow from a convexity analysis of the relative entropy functional H.f /. Indeed, by the convexity of h.z/, it holds that 1 1 1 h.g/ C h0 .g/ .g g/ 2 h.g /: 2

160


Hence, for any large > 0, any times 0 t1 < t2 , and any compact set K R3 , by the non-negativity of h.z/, we have that, for < 1 (so that g > 1), Z t2 Z 1 0 1 h.g/ C h .g/ .g g/ 1fjgjg M dxdvdt 2 t1 KR3 Z t2 Z 1 h.g /M dxdvdt: 2 3 3 t1 R R R z 1 .yz/2 Rz 1 Now, notice that h.z/ D 12 z 2 0 .1Cy/ dy and h0 .z/ D z 0 .1Cy/ 2 2 .yz/ dy, 2 from which we easily deduce the strong convergences 1 1 h.g/1fjgjg ! g 2 1fjgjg and 2 2 1 0 h .g/1fjgjg ! g1fjgjg in L1 .dtdxdv/: Therefore, taking weak limits in the above convexity inequality yields Z t2 Z Z t2 Z 1 2 1 h.g /M dxdvdt g 1fjgjg M dxdvdt lim inf 2 !0 3 3 3 2 t1 KR t1 R R C in .t2 t1 /; which, by monotonicity of the integrands, gives Z t2 Z 1 2 g M dxdvdt C in .t2 t1 /: 3 3 t1 R R 2 Finally, the proof of the lemma is concluded by the arbitrariness of t1 and t2 .

The second assertion of the preceding lemma shows that, in the vanishing limit, the limiting fluctuation belongs to L2 .M dxdv/ uniformly in t. Hence, the weighted L1 -bound implied by the first assertion of Lemma 5.1 is certainly not optimal. Thus, in order to refine our understanding of the limit ! 0, we consider the following renormalized fluctuations q p G D 1 C gO ; Gin D 1 C gO in ; 2 2 q q ˙ ˙ ˙in G D 1 C gO ; G D 1 C gO ˙in ; 2 2 or, equivalently, q

2 p 2 G 1 ; gO in D Gin 1 ; q q 2 2 gO ˙ D G˙ 1 ; gO in˙ D Gin˙ 1 : gO D

(5.3)

5.2 Controls from the entropy dissipation bound

161

Such square-root renormalizations have already been used in previous works on hydrodynamic limits. The advantages of these renormalized fluctuations over the original ones become apparent in the coming lemma, which is, essentially, a modern reformulation of Corollary 3.2 from [11]. Lemma 5.2. Let f .t; x; v/ be a family of measurable, almost everywhere nonnegative distribution functions such that, for all t 0, Z 1 1 H f .t/ D h .g / .t/M dxdv C in : 2 2 R3 R3 Then, as ! of renormalized fluctuations gO is uniformly 0, any subsequence bounded in L1 dtI L2 .M dxdv/ . Proof. The elementary inequality (B.5) implies that, for all t 0, Z Z

2 4 p gO 2 .t/M dxdv D 1 C g 1 .t/M dxdv 2 R3 R3 R3 R3 4 2 H.f /.t/ 4C in ; which is the announced result.

(5.4)

The simple Lemma 5.2 provides important information on any subsequence of fluctuations g . Indeed, a very natural application of this refined a priori estimate follows from decomposing the fluctuations as g D gO C gO 2 2 L1 dtI L2 .Mdxdv/ C L1 dtI L1 .Mdxdv/ : (5.5) 4 Therefore, by Lemma 5.2, the fluctuations g are uniformly bounded in L1 .dtI L2 .Mdxdv//, up to a remainder of order in L1 .dtI L1 .Mdxdv//. In particular, by Lemma 5.1, if g is a weak limit point in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// of a converging subsequence of fluctuations g , then gO also converges towards g in the weak- topology of L1 .dtI L2 .Mdxdv//. As we will see later on, it will be crucial to establish sharper properties of tightness and equi-integrability on the sequence of integrable functions gO 2 . These refinements will follow from the joint control of the fluctuations by the entropy and the entropy dissipation bounds.

5.2 Controls from the entropy dissipation bound Following [11], again, we introduce the non-negative convex function r.z/ D z log.1 C z/; defined over .1; 1/. We may then recast the entropy inequalities (4.29) and (4.36) utilizing this notation to get the entropy dissipation bounds (here, exceptionally, we

162


consider any cross-section b 0, for more generality), for all t 0, Z Z 1 t D.f /.s/ dxds 4 0 R3 2 Z tZ Z 1 q f f b dvdv d dxds C in ; r D 4 G G R3 R3 R3 S2 4 0 and 1 4

Z tZ

C D f C D f C ı2 D fC ; f .s/ dxds R3 0 2 C Z tZ Z 1 q C fC f r b dvdv d dxds D C 4 GC G R3 R3 R3 S2 4 0 2 Z tZ Z 1 q f f r b dvdv d dxds C 4 G 3 3 3 2 4 G R R R S 0 ! Z tZ Z ı2 2 qC; fC f C r b dvdv d dxds C in : C 4 3 3 3 2 2 ıG G R R R S 0

The entropy dissipation bounds are expected to control the size of the collision integrands q , qC , q , q˙ and q , since r.z/ z 2 ;

near z D 0:

However, this behavior only holds asymptotically, as z ! 0, and thus, in order to exploit the entropy dissipation bounds, we will have to rely crucially on Young’s inequality (B.4) for r.z/ and on inequality (B.8), presented in Appendix B. Furthermore, when coupled with a coercivity estimate for some suitable non-singular linearized collision operator, the entropy dissipation bounds will actually provide some control on the relaxation to equilibrium of the fluctuations g and g˙ (see Section 5.3 below). In order to refine our understanding of the limit ! 0, we consider the following renormalized collision integrands

p 2 p 0 0 qO D 2 G G G G ; q q 2 C0 C0 C C C G G G G ; qO D 2 2 p 0 0 p G G G G ; qO D 2 (5.6) q q 2ı 0 ; GC0 G GC G qO C; D 2 q q 2ı C0 C qO ;C D 2 : G0 G G G

5.2 Controls from the entropy dissipation bound

163

The advantages of these renormalized collision integrands over the original ones become apparent in the next lemma. Lemma 5.3. Let fC .t; x; v/ and f .t; x; v/ be two families of measurable, almost everywhere non-negative distribution functions such that, for all t 0, Z Z ı2 t D fC ; f .s/ dxds 4 0 R3 ! Z tZ Z ı2 2 qC; fC f r b dvdv d dxds C in : D 4 ıGC G R3 R3 R3 S2 2 0 Then, as ! 0, any subsequence of renormalized collision integrands qO C; is uniformly bounded in L2 .bMM dtdxdvdv d /. Proof. The elementary inequality (B.8) implies that, for all t 0, Z Z Z C; 2 1 t bMM dvdv d dxds qO 2 0 R3 R3 R3 S2 12 0s Z tZ Z 2 qC; ı2 @ 1C 1A fC f b dvdv d dxds D2 C 4 3 3 3 2 ıG G R R R S 0 Z Z ı2 t D fC ; f .s/ dxds C in ; 4 0 R3 (5.7) as claimed. The simple Lemma 5.3 provides important information on any subsequence of collision integrands qC; . Indeed, a very natural application of this refined a priori estimate follows from decomposing the collision integrands as q 2 C; 2 q qC; D GC G O C; C : (5.8) qO 4ı Hence, at least in the simpler case of the Maxwellian cross-section b 1, we see from Lemma 5.3 that, for any admissible renormalization ˇ.z/, the renormalized collision integrands ˇ 0 GC qC; are uniformly bounded in L1loc .dtdxI L1 .MM dvdv d //, provided the natural entropy and entropy dissipation bounds are satisfied. Moreover, employing Lemma 5.2 and Egorov’s theorem, one can show that C q C 0 ! ˇ 0 .1/ G G in L2loc dtdxI L2 .MM dvdv d / : ˇ G In particular, if q C; is a weak limit point in L2 .MM dtdxdvdv d / of a con verging subsequence of collision integrands qO C; , then ˇ 0 GC qC; converges – up to extraction of jointly converging subsequences – towards ˇ 0 .1/q C; in the weak topology of L1loc dtdxI L1 .MM dvdv d //.

164


5.3 Relaxation towards thermodynamic equilibrium In this section, we establish the relaxation of fluctuations towards thermodynamic equilibrium as a consequence of the relative entropy and the entropy dissipation bounds. As we consider fluctuations around a global equilibrium, the linearized collision operator L, defined in (2.10) and (2.39), is expected to play here a fundamental role. We recall that, for the sake of simplicity, we restrict our attention, in this part of our work, to the case of Maxwellian molecules, that is, to constant collision crosssections, say b 1. Nevertheless, up to additional technical difficulties, the results in this section will be extended to general cross-sections in the remaining parts of this work. The spectral analysis of the linearized collision operator uses crucially the following decomposition based on a clever change of variables sometimes called “Carleman’s collision parametrization”, although it goes back to Hilbert [42] (see equation (17) in [42] and the computations therein). We refer to [70] for more details and to [51] for a modern and general treatment of the linearized Boltzmann operator. Proposition 5.4 (Hilbert decomposition of L). The linearized collision operator L, defined by Z 1 Lg D .Q.Mg; M / C Q.M; Mg// D g C g g 0 g0 M dv d; M R3 S2 can be decomposed as

Lg D g Kg;

where K is a compact integral operator on L2 .Mdv/. As an immediate consequence of the preceding proposition, the operator L satisfies the Fredholm alternative, as well as some coercivity estimate, which will be used to control the relaxation process. We refer to [51] or [70] for details and justifications of the following proposition, or to the proof of the more general Proposition 5.7 below. Proposition 5.5 (Coercivity of L). The linear collision operator L is a non-negative self-adjoint operator on L2 .Mdv/ with nullspace ˚ Ker.L/ D span 1; v1 ; v2 ; v3 ; jvj2 : Moreover, the following coercivity estimate holds: there exists C > 0 such that, for each g 2 Ker.L/? L2 .Mdv/, Z kgk2L2 .M dv/ C g Lg.v/M.v/ dv: R3

5.3 Relaxation towards thermodynamic equilibrium

165

In particular, for any g 2 Ker.L/? L2 .Mdv/, kgkL2 .M dv/ C kLgkL2 .Mdv/ : We will also need the generalization of the preceding propositions to the linearized collision operator for two species of particles L. In fact, employing the results from [51], we easily obtain the following Hilbert decomposition for L. Proposition 5.6 (Hilbert decomposition of L). The linearized collision operator for two species L, defined by Lg C L.g; h/ g ; L D Lh C L.h; g/ h where

L.g; h/ D

1 .Q.Mg; M /CQ.M; M h// D M

can be decomposed as

Z R3 S2

g Ch g 0 h0 M dv d;

g g g L D2 K ; h h h

where K is a compact integral operator on L2 .Mdv/. As an immediate consequence of the preceding proposition, the operator L satisfies the Fredholm alternative, as well as some coercivity estimate, which will be used to control the relaxation process for two species of particles. For the sake of completeness, we provide here a brief justification of the following proposition. Proposition 5.7 (Coercivity of L). The linear collision operator L is a non-negative self-adjoint operator on L2 .Mdv/ with nullspace 2 1 0 v v v jvj Ker.L/ D span ; ; 1 ; 2 ; 3 ; : 0 1 v1 v2 v3 jvj2 Moreover, the following coercivity estimate holds: there exists C > 0 such that, for each gh 2 Ker.L/? L2 .Mdv/, 2 Z g g g L .v/M.v/ dv: C h 2 h h 3 R L .M dv/ In particular, for any

g

2 Ker.L/? L2 .Mdv/, g L g C h 2 h h

L .M dv/

L2 .M dv/

:

166


Proof. The non-negativity and the self-adjointness of L easily follow from a standard use of the collision symmetries by showing that Z gN g L N .v/M.v/ dv h h 3 R Z 1 D g C g g 0 g0 gN C gN gN 0 gN 0 MM dvdv d 4 R3 R3 S2 Z 1 h C h h0 h0 hN C hN hN 0 hN 0 MM dvdv d C 4 R3 R3 S2 Z 1 g C h g 0 h0 gN C hN gN 0 hN 0 MM dvdv d: C 2 R3 R3 S2 (5.9) Next, consider g 2 Ker.L/: h We deduce from (5.9) that, necessarily, g D …g, h D …h and Z 2 … .g h/ .… .g h//0 MM dvdv d D 0: R3 R3 S2

A simple and direct computation shows then that h and g have the same bulk velocity and temperature, which completes the characterization of the kernel of L. In particular, the orthogonal projection onto the kernel of L in L2 .Mdv/ is explicitly given by ! R gCh 1 .g h/M dv C … g 2 : P D 21 RR3 h 2 R3 .g h/M dv C … gCh 2 Finally, since L is positive definite, self-adjoint and satisfies Hilbert’s decomposition from Proposition 5.6, we easily obtain, by the spectral theorem for compact self-adjoint operators, writing g h in the Hilbert basis of eigenvectors of L, that 2 Z g g g g L M dv; C h P h 2 h h R3 L .Mdv/ which concludes the justification of the proposition.

Finally, we also extend the preceding propositions to the linearized collision operator L. In fact, employing the results from [51], we easily obtain the following Hilbert decomposition for L.


167

Proposition 5.8 (Hilbert decomposition of L). The linearized collision operator L, defined by 1 Lg D L.g; g/ D .Q.Mg; M / Q.M; Mg// M Z g g g 0 C g0 M dv d; D R3 S2

can be decomposed as

Lg D g Kg;

where K is a compact integral operator on L2 .Mdv/. Note that the definition of L above coincides with (2.66). As an immediate consequence of the preceding proposition, the operator L satisfies the Fredholm alternative, as well as some coercivity estimate, which will be used to control the relaxation process for two species of particles. For the sake of completeness, we provide here a brief justification of the following proposition. Proposition 5.9 (Coercivity of L). The linear collision operator L is a non-negative self-adjoint operator on L2 .Mdv/ with nullspace Ker.L/ D span f1g : Moreover, the following coercivity estimate holds: there exists C > 0 such that, for each g 2 Ker.L/? L2 .Mdv/, Z gLg.v/M.v/ dv: kgk2L2 .M dv/ C R3

In particular, for any g 2 Ker.L/? L2 .Mdv/, kgkL2 .M dv/ C kLgkL2 .Mdv/ : Proof. The non-negativity and self-adjointness of L easily follow from a standard use of the collision symmetries by showing that Z gLh.v/M.v/ dv R3 Z 1 g g g 0 C g0 h h h0 C h0 MM dvdv d: D 4 R3 R3 S2 (5.10) Next, consider g 2 Ker.L/. We deduce from (5.10) that, necessarily, g g D g 0 g0

168


almost everywhere. Hence, since the change of variable 7! merely exchanges v 0 and v0 , we find, averaging over 2 S2 , that Z 0 1 g g0 d D 0; g g D 2 jS j S2 for every v; v 2 R3 . It follows that g is a constant function. Finally, since L is positive definite, self-adjoint and admits the Hilbert decomposition from Proposition 5.8, we easily obtain, by the spectral theorem for compact self-adjoint operators, writing g in the Hilbert basis of eigenvectors of L, that 2 Z Z g g M dv C gLgM dv; R3

L2 .Mdv/

R3

which concludes the justification of the proposition.

It is to be emphasized that, since we are only considering here the case of Maxwellian b 1, the linearized operator L can be explicitly rewritten, R molecules using that S2 g 0 g0 d D 0, as Z ˇ ˇ Lg D ˇS2 ˇ g g M dv ; R3

which renders the proofs of Propositions 5.8 and 5.9 trivial. However, we chose to provide more robust justifications of both propositions, which work in more general settings of hard and soft potentials as well.

5.3.1 Infinitesimal Maxwellians Using the usual relative entropy and entropy dissipation bounds together with the coercivity of the linearized collision operator, we easily get that each species of particles reaches almost instantaneously the local thermodynamic equilibrium in the fast relaxation limit. More precisely, we have the following lemma. Lemma 5.10. Let f .t; x; v/ be a family of measurable, almost everywhere nonnegative distribution functions such that, for all t 0, Z Z 1 t 1 H f D .f / .s/ dxds C in : .t/ C 2 4 0 R3 Then, as ! 0, any subsequence of renormalized fluctuations gO satisfies the relaxation estimate kgO …gO kL2 .Mdv/ O./ kgO k2L2 .Mdv/ C O ./L2 .dt dx/ ; where … denotes the orthogonal projection on Ker L in L2 .Mdv/.

(5.11)


169

Proof. We start from the elementary decomposition p 2 p LgO D Q .gO ; gO / Q G ; G ; (5.12) 2 and we estimate each term in the right-hand side separately. First, since b 1, it is readily seen that the quadratic collision operator is continuous on L2 .Mdv/: Z 0 0 gO gO gO gO M dv d kQ.gO ; gO /kL2 .Mdv/ D R3 S2

L2 .Mdv/

Z 12 ˇ 2 ˇ 12 0 0 2 ˇS ˇ gO gO gO gO M dv d R3 S2 2 L .Mdv/ ˇ 2ˇ 2 ˇS ˇ kgO k2L2 .Mdv/ ;

which, when combined with the bound (5.4) from the proof of Lemma 5.2, yields ˇ ˇ 8 ˇS2 ˇ C in : kQ.gO ; gO /k L1 dxIL2 .Mdv/

Furthermore, employing the uniform L2 -estimate (5.7) from the proof of Lemma 5.3 on the renormalized collision integrands qO and the Cauchy–Schwarz inequality, we deduce that Z tZ Z p 2 1 p Q G ; G M dvdxds 2 R3 R3 0 2 Z Z Z Z 1 t qO M dv d M dvdxds D 3 3 3 2 4 ˇ 2 ˇ0 Z R Z R Z R S ˇS ˇ t .qO /2 MM dvdv d dxds 4 0 R3 R3 R3 S2 ˇ 2ˇ Z Z ˇS ˇ t ˇ ˇ D.f /.s/ dxds ˇS2 ˇ C in : 4 R3 0 Therefore, combining (5.12) with the coercivity estimate from Proposition 5.5 leads to kgO …gO kL2 .Mdv/ C kLgO kL2 .Mdv/

p 1 p C kQ.gO ; gO /kL2 .Mdv/ C 2 Q G ; G

p 1 p C kgO k2L2 .Mdv/ C Q G ; G 2

! L2 .Mdv/

!

L2 .Mdv/

D C kgO k2L2 .Mdv/ C O ./L2 .dt dx/ ; which concludes the proof of the lemma.

170


5.3.2 Bulk velocity and temperature When considering the two-species Vlasov–Maxwell–Boltzmann system (4.35), we have an additional relaxation estimate on bulk velocities and temperatures coming from the mixed entropy dissipation. Lemma 5.11. Let fC .t; x; v/ and f .t; x; v/ be two families of measurable, almost everywhere non-negative distribution functions such that, for all t 0, 1 1 C H f C 2 H f 2 Z Z C 1 t C 4 D f C D f C ı2 D fC ; f .s/ dxds C in : 0 R3 Then, as ! 0, any subsequence of renormalized fluctuations gO ˙ satisfies the relaxation estimate 2 C gO C gO C P gO ; O./ C O gO 2 gO gO L2 .Mdv/ ı L2loc dt IL2 .dx/ L .Mdv/ (5.13) where P denotes the orthogonal projection on Ker L in L2 .Mdv/. In particular, further considering the densities O˙ , bulk velocities uO ˙ and temperatures O˙ , respectively associated with the renormalized fluctuations gO ˙ , it holds that

ı C hO D gO gO nO where nO D OC O , and

is uniformly bounded in L1loc dtdxI L2 .Mdv/ ; (5.14)

ı C ı O C O D and w O uO uO jO D are uniformly bounded in L1loc .dtdx/: Finally, one also has the refined relaxation estimate hO ı nO gO ˙ O˙ 2 L2 .Mdv/ C ˙ gO 2 O.ı/ gO gO nO 2 C O .1/

L .Mdv/

L .Mdv/

:

(5.15)

2 L2 loc dt IL .dx/

Proof. First, a direct application of Lemma 5.10 yields ˙ 2 ˙ O 2 gO …gO ˙ 2 L .Mdv/ O./ g L .Mdv/ C O ./L2 dt IL2 .dx/ : loc

(5.16)

171


Next, we apply similar arguments from the proof of Lemma 5.10 to the mixed entropy dissipation D fC ; f . Thus, according to the definitions of L.g; h/ and Q.g; h/ in (2.39), we start from the elementary decomposition q q ˙ ˙ 2 ˙ (5.17) L gO ; gO D Q gO ; gO Q G ; G ; 2 and we estimate each term in the right-hand side separately. Since b 1, it is readily seen that the quadratic collision operator is continuous on L2 .Mdv/: Z ˙ ˙0 0 ˙ Q gO ; gO 2 D gO gO gO gO M dv d L .Mdv/ R3 S2

Z 12 ˇ 2 ˇ 12 ˙0 0 2 ˇS ˇ M dv d gO gO gO ˙ gO R3 S2 2 L .Mdv/ ˇ 2 ˇ C ˇ ˇ 2 S gO L2 .Mdv/ kgO kL2 .Mdv/ ;

L2 .Mdv/

(5.18) which, when combined with the bound (5.4) from the proof of Lemma 5.2, yields ˇ ˇ ˙ 8 ˇS2 ˇ C in : Q gO ; gO L1 dxIL2 .Mdv/ Furthermore, employing the uniform L2 -estimate (5.7) from the proof of Lemma 5.3 on the renormalized collision integrands qO ˙ and the Cauchy–Schwarz inequality, we deduce that q 2 Z tZ Z p ı C Q G ; G M dvdxds 2 R3 R3 0 2 Z Z Z Z 1 t C; D qO M dv d M dvdxds 3 3 3 2 4 (5.19) ˇ 2 ˇ0 Z R Z R Z R S ˇS ˇ t C; 2 MM dvdv d dxds qO 3 3 3 2 4 ˇ 2ˇ ˇ 2 ˇ 20 Z R Z R R S t ˇS ˇ ˇS ˇ ı C D f ; f .s/ dxds C in : 2 4 2 R3 0 Notice that the same estimate holds on the renormalized collision integrands qO ;C , which yields q 2 p ı Q G ; GC M dvdxds 2 3 3 R 0 ˇ 2 ˇR 2 Z Z ˇ 2ˇ t ˇS ˇ ı ˇS ˇ C D f ; f .s/ dxds C in : 4 3 2 2 R 0

Z tZ

Z

(5.20)

172


Therefore, combining (5.16) and (5.17) with the coercivity estimate from Proposition 5.7 leads to C gO C P gO gO gO L2 .Mdv/ C LgO C L gO ; gO gO C C C L C C C gO LgO L2 .Mdv/ L gO ; gO L2 .Mdv/ L2 .Mdv/ C gO C …gO C Q gO ; gO C C C gO …gO 2 Q gO ; gO C 2 L .Mdv/ L .Mdv/ 0 p

1 p C ı @Q G ; G A p CC p 2 ı Q G ; GC 2 L .Mdv/ 2 gO C CO ; O./ C O./ 2 gO 2 L2 loc dt IL .dx/ ı L2loc dt IL2 .dx/ L .Mdv/ which concludes the proof of the relaxation estimate (5.13). Then, in order to deduce the control of hO , jO and wO , it suffices to notice that C C 1 gO gO P D gO C gO OC O ; 1 gO gO and

! R C C gO C gO … 2 12 R3 gO C gO M dv gO gO … P D R gO gO C gO gO … 2 12 R3 gO gO C M dv 0 C

1 uO uO OC O jvj2 3 v C 2 2 2 A 2

; D @ uO uO C O OC jvj2 3 v C 2 2 2 2

whence C 2 C … gO P gO 2 gO gO

L .Mdv/

D

2 2 3 C 1 C O O : C uO uO 2 4

There only remains to establish the more precise relaxation estimate (5.15) on hO , which is achieved by employing the coercivity of the operator L. To this end, we use the identities (5.17) to decompose ı LhO D L gO C gO q q p p 2ı ı C Q gO ; gO Q gO ; gO C 2 Q GC ; G Q G ; GC D 2 Z

C; ı C ˙ ˙ C Q gO gO ; gO Q gO ; gO gO qO qO ;C M dv d: D 2 R3 S2


173

It follows that

i h O ı L hO nO gO ˙ D Q h ; gO ˙ Q gO ˙ ; hO 2 2 Z C; qO qO;C M dv d: R3 S2

Therefore, repeating the estimates (5.18), (5.19) and (5.20), we find that ˙ O gO 2 L hO ı nO gO ˙ O./ h 2 L .Mdv/ L .Mdv/ 2 2 L .Mdv/ C O .1/ 2 L2 loc dt IL .dx/

D O.ı/ gO C gO nO L2 .Mdv/ gO ˙ L2 .Mdv/ : C O .1/ 2 L2 loc dt IL .dx/

Finally, employing the coercivity of L from Proposition 5.9, we easily conclude that (5.15) holds, which completes the proof of the lemma. Remark. Under the hypotheses of the preceding lemma, it is possible to obtain a very explicit identity providing some improved information on the relaxation of hO . To this end, we decompose Z C; qO qO ;C M dv d R3 S2 q q p p 2ı 2ı C C D 2Q G ; G 2 Q G ; G q q q p 2ı nO C ˙ ˙ D 2Q G G G ; G 2 1 C 2 O˙ q q q p 2ı nO C ˙ ˙ 2Q G ; G G G 2 1 C 2 O˙ q q ı gO ˙ O˙ ı gO ˙ O˙ G˙ ; hO nO ˙ : D Q hO nO ˙ ; G˙ Q 2 1 C 2 O 2 1 C 2 O Then, since we are only considering here the Maxwellian cross-section b 1, notice that the gain terms cancel each other q q ı gO ˙ O˙ ı gO ˙ O˙ C C ˙ ˙ O O D 0; Q h nO ; G Q G ; h nO 2 1 C 2 O˙ 2 1 C 2 O˙ for the change of variable 7! merely exchanges v 0 and v0 , and that one of the two loss terms vanishes: q ı gO ˙ O˙ ˙ O D 0: Q G ; h nO 2 1 C 2 O˙

174


Thus, on the whole, we are left with the identity

˙ ˙ q ˇ 2ˇ ˇS ˇ 1 C O˙ hO ı nO gO ˙ O˙ D Q hO ı nO gO O ; G ˙ 2 2 2 1 C 2 O˙ Z C; D qO qO ;C M dv d; R3 S2

which yields the control, in view of Lemma 5.3,

ı 1 C O˙ hO nO gO ˙ O˙ D O.1/L2.Mdt dxdv/ : 2 2 2

In particular, further integrating against vMdv and jvj3 1 Mdv, we obtain that 1 C O˙ jO 2 ˙ 1 C O wO 2

ı nO uO ˙ D O.1/L2 .dt dx/ ; 2 ı O˙ nO D O.1/L2 .dt dx/ : 2

These estimates are slightly more precise than (5.15). However, their significance is unclear.

5.4 Improved integrability in velocity Another important consequence of the control on the relaxation is to provide further integrability on the renormalized fluctuations gO and gO ˙ with respect to the v variable at infinity. The following result, which was first established as such in [70] (see Lemma 3.2.5 therein) and [38] (see Proposition 3.2 therein), improves Lemma 5.2 and is a direct consequence of Lemma 5.10. It constitutes a significant simplification with respect to earlier works on hydrodynamic limits of the Boltzmann equation, which required convoluted estimates to establish some improved integrability in the v variable. Lemma 5.12. Let f .t; x; v/ be a family of measurable, almost everywhere nonnegative distribution functions such that, for all t 0, Z Z 1 1 t H f .t/ C 4 D .f / .s/ dxds C in : 2 0 R3 Then, as ! fluctuations gO is uniformly 0, any subsequence of renormalized bounded in L2loc dtdxI L2 1 C jvj2 Mdv .

5.4 Improved integrability in velocity

175

Furthermore, the family jgO j2 is equi-integrable in v (or uniformly integrable in v) in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if A K is a measurable set satisfying Z sup 1A .t; x; v/ dv < ; .t;x/2Œ0;1/R3

R3

Z

then

jgO j2 dtdxdv < :

sup >0

A

We also have that, for any > 0 and any 1 p < 2, the families 1fjgO j1g jgO j2 and

jgO p j2 1C G

are uniformly bounded in L1loc dtdxI Lp .Mdv/ .

Proof. The crucial idea behind these results rests upon decomposing gO according to gO D .gO …gO / C …gO ;

(5.21)

and then using the control on the relaxation provided by Lemma 5.10: gO …gO D O./L1 dt dxIL2 .Mdv/ : loc

2 First, we establish the uniform control on the high speed tails 2of jgO j , i.e., the 2 2 uniform weighted integrability estimate in Lloc dtdxI L 1 C jvj Mdv . To this end, we start from the decomposition

.1 C jvj/2 jgO j2 D gO .1 C jvj/2 …gO C Œ.1 C jvj/gO Œ.1 C jvj/ .gO …gO / : (5.22) Next, recalling from Lemma 5.2 that gO is uniformly bounded in L1 dtI L2 .Mdxdv/ ;

we see, by the definition of the hydrodynamic projection …, for any 1 p < 1, that (5.23) .1 C jvj/2 …gO is uniformly bounded in L1 dtI L2 dxI Lp .Mdv/ ; whence, for any 1 r < 2,

gO .1 C jvj/2 …gO is uniformly bounded in L1 dtI L1 dxI Lr .Mdv/ ; (5.24) which takes care of the first term in right-hand side of (5.22).

176


In order to estimate the second term in the right-hand side of (5.22), we first apply 2 , ˛ D 4 , ˇ D 4 and > 0, to Young’s inequality (B.3) with z D g , y D .1Cjvj/ get, employing the elementary inequality (B.7), that 4 4 .1 C jvj/2 1j D jg jG j 2 2 4 .1Cjvj/ 4 2 h .g / C 2 e : Therefore, setting D 4 in (5.25), we obtain that .1 C jvj/2 gO 2 .1 C jvj/2

(5.25)

j.1 C jvj/gO .1 C jvj/ .gO …gO / j 4 .1Cjvj/2 4p h .g /.1 C jvj/ jgO …gO j C e 8 .1 C jvj/ jgO …gO j 1 4 .1Cjvj/2 16 2 h .g / C .1 C jvj/2 jgO …gO j2 C e 8 .1 C jvj/ jgO …gO j 4

4 .1Cjvj/2 1 16 2 2 2 2 h .g / C .1 C jvj/ jgO j C j…gO j C e 8 .1 C jvj/ jgO …gO j ; 2 which, by virtue of the uniform entropy bound, the uniform estimate (5.23), and the relaxation estimate (5.11) from Lemma 5.10, yields 1 j.1 C jvj/gO .1 C jvj/ .gO …gO / j O.1/L1 dt dxIL1 .Mdv/ C .1 C jvj/2 jgO j2 : 2 loc (5.26) On the whole, incorporating (5.24) and (5.26) into the decomposition (5.22), we deduce that 1 .1 C jvj/2 jgO j2 O.1/L1 dt dxIL1 .Mdv/ C .1 C jvj/2 jgO j2 : 2 loc Hence, .1 C jvj/2 jgO j2 D O.1/L1 dt dxIL1 .Mdv/ ; loc

which is the expected result. We establish now the uniform integrability statement of the lemma. To this end, we start from the decomposition, for any large > e, jgO j2 D 1fG >g jgO j2 C 1fG g gO …gO C 1fG g gO .gO …gO / :

(5.27)

Then, we use the relative entropy bound, a pointwise estimate of h.z/, for z > , and the elementary inequality (B.7) to control the large tails of G as follows 4 8 1fG >g jgO j2 2 1fG >g jG 1j 2 1fG >g G 8 8 1fG >g G .log G 1/ 2 h .g / 2 .log 1/ .log 1/ 1 DO ; log L1 dt IL1 .Mdxdv/ (5.28)


177

which takes care of the first term in the right-hand side of (5.27), while the second term is handled by estimate (5.24). As for the remaining term in (5.27), we deduce from the relaxation p (5.11) in Lemma 5.10 and from the pointwise estimate ˇ ˇ estimate ˇ1fG g gO ˇ 2 1 C , which follows straightforwardly from (B.7), that p : 1fG g gO .gO …gO / D O

1 2 Lloc dt dxIL .Mdv/

Thus, on the whole, we have established from the decomposition (5.27) that, for any arbitrarily large and each 1 r < 2, p 1 2 ;

1 jgO j D O CO Lloc dt dxILr .Mdv/ log L1 dt IL1 .Mdxdv/ which clearly implies that jgO j2 is locally uniformly integrable in v. The final statement is easily obtained by combining decomposition (5.21) with the bounds gO D O.1/L1.dt IL2 .Mdxdv// ; 1 C jgO j gO 1 DO ; 1 C jgO j

L1 .dt dxdv/ p for any > 0, and noticing that 1 C jgO j max f2; 1 C 2 g 1 C G . We indeed find, for any 1 p < 2, that gO gO gO 2 C …gO D .gO …gO / 1 C jgO j p 1 C jgO j 1 C jgO j p

L .Mdv/

L .Mdv/

1 kgO …gO kLp .Mdv/ C kgO …gO kLp .Mdv/

C kgO …gO kL2 .Mdv/ C C kgO k2L2 .Mdv/

C kgO k2L2 .Mdv/ C kqO kL2 MM dvdv d D O.1/L1

C O.1/

dt IL1 .dx/

L2 .dt dx/ ;

which concludes the proof of the lemma.

In the two-species case, the preceding lemma has simple but important consequences on the integrability of the difference of fluctuations, which is the content of the next lemmas. Lemma 5.13. Let fC .t; x; v/ and f .t; x; v/ be two families of measurable, almost everywhere non-negative distribution functions such that, for all t 0, 1 C 1 H f C 2 H f 2 Z Z C 1 t C 4 D f C D f C ı2 D fC ; f .s/ dxds C in : 0 R3

178


Then, as ! 0, considering the densities ˙ , bulk velocities u˙ and temperatures ˙ associated with any subsequence of fluctuations g˙ , it holds that

ı C g g n is uniformly bounded in L1loc dtdxI L1 1 C jvj2 Mdv ; h D

(5.29)

where n D C , and j D

ı C ı C and w D u u 1 are uniformly bounded in Lloc .dtdx/:

Proof. According to the decomposition (5.5), it is readily seen that Z

ˇ C ˇ2 ı ˇˇ C ˇˇ2 ˇgO ˇ jgO j2 M dv ; h D hO C gO jgO j2 C 4 R3 whence, by virtue of Lemma 5.12, h D hO C O.ı/L1 dt dxIL1 .1Cjvj2 /Mdv ; loc

which establishes the uniform bound on h , thanks to the uniform bound on hO from Lemma 5.11. 2 Finally, integrating the above decomposition against vMdv and jvj3 1 Mdv clearly yields j D jO C O.ı/L1

loc .dt dx/

and

w D wO C O.ı/L1

loc .dt dx/

;

which, employing the uniform bounds on jO and wO from Lemma 5.11, concludes the justification of the lemma. Lemma 5.14. Let fC .t; x; v/ and f .t; x; v/ be two families of measurable, almost everywhere non-negative distribution functions such that, for all t 0, 1 C 1 H f C 2 H f 2 Z Z C 1 t C 4 D f C D f C ı2 D fC ; f .s/ dxds C in : 0 R3 Then, as ! 0, in the case of weak interspecies interactions, i.e., when ı D o.1/ and ı is unbounded, any subsequences of fluctuations g˙ and renormalized fluctuations gO ˙ satisfy that

ı C ı C g g n gO gO nO and hO D are weakly relatively compact in L1loc dtdxI L1 1 C jvj2 Mdv ; h D


and

179

ı C ı C u u ; ; w D C ı ı O C O D ; w O uO uO ; jO D are weakly relatively compact in L1loc .dtdx/: j D

Moreover, as ! 0, in the case of strong interspecies interactions, i.e., when ı D 1, any subsequences of fluctuations g˙ and renormalized fluctuations gO ˙ satisfy that hO C 1 C gO gO

and

is uniformly bounded in L2loc dtI L2 .Mdxdv/ ; L2 .Mdv/

jO C 1 C gO gO L2 .Mdv/

wO C 1 C gO gO L2 .Mdv/ are uniformly bounded in L2loc dtI L2 .dx/ ; and

while h 1 C gO C gO L2 .Mdv/ is uniformly bounded in L2loc dtdxI L1 ..1 C jvj/Mdv/ and weakly relatively compact in L1loc dtdxI L1 ..1 C jvj/Mdv/ ; and

j C 1 C gO gO

is uniformly bounded in L2loc .dtdx/:

L2 .Mdv/

Proof. We handle the case ı D o.1/ first. We have already established the uniform boundedness of hO , jO and wO in Lemma 5.11, while the uniform boundedness of h , j and w comes from Lemma 5.13. Moreover, the tightness in v of hO is easily deduced from the bound (5.14) from Lemma 5.11. Therefore, according to the Dunford–Pettis compactness criterion (see [68]), it suffices to show that h , j , w , hO , jO and wO are uniformly integrable in all variables and that h is tight in v. We deal with hO , jO and wO first. To this end, simply notice that Lemma 5.11 provides the control 2 ˇ ˇ gO C ˇjO ˇ C jwO j C hO 2 O.ı/ C O .1/L2 dt IL2 .dx/ ; gO 2 L .Mdv/ loc L .Mdv/

180


whence, by Lemma 5.2, ˇ ˇ ˇjO ˇ C jwO j C hO

L2 .Mdv/

O.ı/L1 dt IL1 .dx/ C O .1/L2 dt IL2 .dx/ ; loc

which establishes the equi-integrability of jO , wO and khO kL2 .Mdv/ in t and x. Furthermore, since hO is clearly equi-integrable in v thanks to the bound (5.14) from Lemma 5.11, a direct application of Lemma 5.2 from [37] yields that hO is equiintegrable in all variables. Next, we deduce the relative weak compactness of h , j and w from the relative weak compactness of hO , jO and wO employing the decomposition (5.5), which clearly yields Z

ˇ C ˇ2 ı ˇˇ C ˇˇ2 ˇgO ˇ jgO j2 M dv : h D hO C gO jgO j2 4 R3 Therefore, since ı D o.1/, it is readily seen, by virtue of the uniform integrability of jO , wO and hO and the uniform boundedness of gO ˙ in L2loc .dtdxI L2 ..1Cjvj2 /Mdv// from Lemma 5.12, that j , w and h are uniformly integrable in all variables as well, and that h is tight in v in L1loc .dtdxI L1 ..1 C jvj2 /Mdv//. We turn now to the case ı D 1. It is readily seen that the estimate (5.15) from Lemma 5.11 provides the refined control hO 2 C gO C gO L2 .Mdv/ gO ˙ L2 .Mdv/ C O .1/L2 dt IL2 .dx/ : L .Mdv/ loc

In particular, in view of the boundedness of gO ˙ in L1 .dtI L2 .Mdxdv//, Lemma 5.2 shows that hO C 1 C gO gO L2 .Mdv/ and, incidentally, that ˇ ˇ ˇjO ˇ C jwO j C 1 C gO gO 2

is uniformly bounded in L2loc dtI L2 .Mdxdv/ ;

is uniformly bounded in L2loc dtI L2 .dx/ :

L .Mdv/

Next, we deduce the relative weak compactness of h and the uniform bound on j from the uniform bound on hO employing the decomposition (5.5), which clearly yields Z C C C 1 C O gO gO gO C gO h D h C gO gO gO C gO M dv : (5.30) 4 R3


181

Therefore, it is readily seen that kh kL1 ..1Cjvj/Mdv/ C hO L2 .Mdv/ C C gO C gO L2 .Mdv/ gO C C gO L2 .1Cjvj2 /Mdv ; and so is uniformly bounded in L2loc dtdxI L1 ..1Cjvj/Mdv/ ;

h C 1 C gO gO L2 .Mdv/ and, incidentally, j C 1 C gO gO

is uniformly bounded in L2loc .dtdx/:

L2 .Mdv/

Now, it is readily seen from the uniform bound on hO in L1loc .dtdxI L2 .Mdv// established in Lemma 5.11 and from the equi-integrability in v of the families jgO ˙ j2 established in Lemma 5.12 that the decomposition (5.30) yields that h is equiintegrable in v as well. Consequently, a direct application of Lemma 5.2 from [37] yields that the family h C 1 C gO gO 2

L .Mdv/

is equi-integrable in all variables, which, according to the Dunford–Pettis compactness criterion (see [68]), implies its weak relative compactness in L1loc .dtdxI L1 .Mdv//. Finally, using the uniform bound (5.29) from Lemma 5.13, we deduce that the family h C 1 C gO gO 2

L .Mdv/

is weakly relatively compact in L1loc .dtdxI L1 ..1 C jvj/Mdv//, which concludes the proof of the lemma.

Chapter 6

Lower-order linear constraint equations and energy inequalities In the preceding chapter, we have established uniform estimates and controls on the fluctuations and collision integrands by analyzing the relative entropy and entropy dissipation bounds. At this stage, we have now all the necessary tools to derive the asymptotic lower-order linear constraint equations and energy inequalities from Theorems 4.5 and 4.6. This first part of the rigorous convergence proofs is therefore very similar for both theorems. The derivation of higher-order and nonlinear constraint equations – in particular, constraints pertaining to Theorem 4.7 – is performed in Chapter 8 and will require more advanced methods and refined properties on the fluctuations. More precisely, strong compactness and nonlinear weak compactness properties of the fluctuations, established later on in Chapter 7, will allow us to obtain the remaining constraint equations such as Ohm’s law.

6.1 Macroscopic constraint equations for one species The macroscopic constraint equations are obtained by integrating the limiting kinetic equation against the collision invariants. In the simplest case of a one-species plasma, taking limits in the kinetic equation is straightforward once we introduce the suitable renormalization. Proposition 6.1. Let .f ; E ; B / be the sequence of renormalized solutions to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) considered in Theorem 4.5. In accordance with Lemmas 5.1, 5.2 and 5.3, denote by q 2 L2 MM dtdxdvdv d ; g 2 L1 dtI L2 .Mdxdv/ ; and E; B 2 L1 dtI L2 .dx/ ; any joint limit points of the families gO and qO defined by (5.3) and (5.6), E and B , respectively. Then, one has Z qM dv d D W rx u C rx ; (6.1) R3 S2

where u and are, respectively, the bulk velocity and temperature associated with the limiting fluctuation g, and and are the kinetic fluxes defined by (2.14).

184

6 Lower-order linear constraint equations and energy. . .

Furthermore, , u, and E satisfy the following constraints div u D 0;

rx . C / E D 0;

(6.2)

where is the density associated with the limiting fluctuation g. Proof. We start from some square root renormalization of the scaled Vlasov–Boltzmann equation (4.28). More precisely, we choose the admissible renormalization p z C a 1 ˇ.z/ D ; for some given 1 < a < 4, which yields, using the decomposition of collision integrands (5.8), p G G C a 1 .@t C v rx C .E C v ^ B / rv / E v p 2 G C a p Z p G D p G qO M dv d 2 G C a R3 S2 (6.3) Z 2 qO 2 M dv d C p 8 G C a R3 S2 def

D Q1 C Q2 :

Then, thanks to Lemma 5.2, it holds that p G D 1 C O./L2 dt IL2 M loc

dxdv

;

whence, employing the uniform bound qO 2 L2 .MM dtdxdvdv d / from Lemma 5.3, p Z G 1 Q D p qO M dv d C O./L1 dt dxIL2 .Mdv/ ; loc 2 G C a R3 S2 (6.4)

a 2 2 2 Q D O : 1 L

Mdt dxdv

Next, since, decomposing according to the tails of G , p p ˇ ˇ ˇ G C a 1 a1 G 1 ˇˇ ˇ Dp a p ˇ ˇ C G C G a1 a p 1fG > 1 g C 2 1 1fG 1 g p a 2 2 C G C G p ˇ ˇ p ˇ G 1 ˇ a ˇ; O a1 L1 .dt dxdv/ C 2 2 C 2 ˇˇ ˇ

6.1 Macroscopic constraint equations for one species

185

one proves, by using Lemma 5.2 or Lemma 5.12, that p a G C a 1 ; gO D O 2 1 1 2 L .dt dxdv/ p a G C a 1 ; gO D O a1 L1 .dt dxdv/ C O 2 1 2 L dt IL2 .Mdxdv/ p a G C a 1 ; 2 gO D O a1 L1 .dt dxdv/ C O 2 2 Lloc dt dxIL2 1Cjvj2 Mdv (6.5) and G E v p G C a p a G C a 1 D E v 1 C E v p G C a

a a C O a C 2 : D E v C O C 2 C1 1 1 1 2 dt IL .Mdxdv/

L

L

dt IL .Mdxdv/

In particular, employing (6.5) to deduce that ˇ ˇp ˇ ˇ p ˇ ˇ G C a 1 1 ˇ ˇ ˇD p ˇ ˇ ˇ p G 1 g O ˇ ˇ G C a 2 ˇ G C a ˇ ˇp ˇ ˇ G C a 1 1 ˇ 1 a ˇ 2ˇ gO ˇˇ 2 a C O ./L2 dt dxIL2 1Cjvj2 Mdv ; O 2 1 L

.dt dxdv/

loc

we obtain the following refinement of (6.4): Z a 1 1 Q D qO M dv d C O 2 2 L .Mdt dxdv/ 2 R3 S2 ; C O./

1 L1 loc dt dxIL ..1Cjvj/Mdv/ a

Q2 D O 2 2

L1 .Mdt dxdv/

:

Therefore, taking weak limits in (6.3) leads to Z qM dv d; v rx g E v D

(6.6)

R3 S2

which, together with the fact that, according to Lemma 5.10, g is an infinitesimal Maxwellian, provides that Z jvj2 qM dv d D div . C /v C u C u C E v 3 R3 S2 1 D . W rx u C rx / C .rx . C / E/ v C .div u/ jvj2 : 3

186


Then, remarking that q inherits the collisional symmetries of q and qO , we get 0 1 Z 1 @ A MM dvdv d D 0; v q R3 R3 S2

jvj2 2

so that, since .v/ and .v/ are orthogonal to the collisional invariants, the constraints (6.2) hold. The proof of the proposition is complete.

6.2 Macroscopic constraint equations for two species, weak interactions In the case of a two-species plasma, the renormalization process is more complicated because there are two different distributions. Nevertheless, for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded, we have a result quite similar to the preceding proposition. As for strong interspecies interactions, i.e., ı D 1, even the lowest-order constraints will require dealing with nonlinear terms and, therefore, will be handled with more advanced techniques in Chapter 8 (see Propositions 8.3 and 8.4). Proposition 6.2. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.6 for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded. In accordance with Lemmas 5.1, 5.2 and 5.3, denote by g ˙ 2 L1 dtI L2 .Mdxdv/ ; q ˙ ; q ˙; 2 L2 .MM dtdxdvdv d /; and E; B 2 L1 dtI L2 .dx/ any joint limit points of the families gO ˙ , qO ˙ and qO ˙; defined by (5.3) and (5.6), E and B , respectively. Then, one has Z q ˙ M dv d D W rx u C rx ; (6.7) R3 S2

where u and are, respectively, the bulk velocity and temperature associated with the limiting fluctuations g ˙ , and and are the kinetic fluxes defined by (2.14). Furthermore, ˙ , u and satisfy the constraints div u D 0; rx ˙ C D 0; (6.8) where ˙ are the densities respectively associated with the limiting fluctuations g ˙ . In particular, ˙ C D 0 holds and, moreover, since C the strong Boussinesq relation C rx D 0, it also holds that D .

6.2 Macroscopic constraint equations for two species. . .

187

Proof. We start from some square root renormalization of the scaled Vlasov–Boltzmann equation (4.35). More precisely, we choose, as previously in the proof of Proposition 6.1, the admissible renormalization p z C a 1 ˇ.z/ D ; for some given 1 < a < 4, which yields, recalling the definitions (5.6) of renormalized collision integrands and using the decomposition of collision integrands (5.8), p G˙ G˙ C a 1 ıE v p .@t C v rx ˙ ı .E C v ^ B / rv / 2 G˙ C a p Z q G˙ ˙q D p G O ˙ M dv d ˙ a 3 2 2 G C R S Z ˙ 2 2 C p qO M dv d 8 G˙ C a R3 S2 p Z q ı G˙ ˙; C p G qO M dv d 2 G˙ C a R3 S2 Z ˙; 2 2 C p M dv d: qO ˙ a 8 G C R3 S2 (6.9) The proof follows then the exact same lines as the proof of Proposition 6.1. In particular, we obtain without any additional difficulty the fact that the right-hand side of (6.9) converges weakly to Z 1 q ˙ M dv d; 2 R3 S2 while the renormalized densities satisfy, following (6.5), p a G˙ C a 1 ; gO ˙ D O 2 1 1 2 L .dt dxdv/ p a G˙ C a 1 ; 2 gO ˙ D O a1 L1 .dt dxdv/ C O 2 1 L dt IL2 .Mdxdv/ p a G˙ C a 1 : 2 gO ˙ D O a1 L1 .dt dxdv/ C O 2 2 Lloc dt dxIL2 1Cjvj2 Mdv Next, using the uniform L1 .dtI L2 .dx// bounds on E and B , as well as the bounds on the renormalized fluctuations p G˙ G˙ C a 1 ; p and 2 G˙ C a

L2loc .dtdxI L2 .Mdv//

188


we easily obtain that all the terms coming from the Lorentz force in (6.9) vanish in the weak limit: p G˙ G˙ C a 1 ˙ı .E C v ^ B / rv ! 0: ıE v p 2 G˙ C a Therefore, taking weak limits in (6.9) leads to Z v rx g ˙ D q ˙ M dv d;

(6.10)

R3 S2

which, together with the fact that, according to Lemmas 5.10 and 5.11, g C and g are infinitesimal Maxwellians, which differ only by their densities C and , shows that Z jvj2 u C u C q ˙ M dv d D div .˙ C /v C 3 R3 S2 1 D . W rx u C rx / C rx .˙ C / v C .div u/ jvj2 : 3 Then, remarking that q ˙ inherits the collisional symmetries of q˙ and qO˙ , we get 0 1 Z 1 q ˙ @ v A MM dvdv d D 0; R3 R3 S2

jvj2 2

so that, since .v/ and .v/ are orthogonal to the collisional invariants, the constraints (6.8) hold. The proof of the proposition is complete. ˙ Proposition 6.3. Let f ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.6 for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded. In accordance with Lemmas 5.1, 5.2, 5.3, 5.13 and 5.14 denote by g ˙ 2 L1 dtI L2 .Mdxdv/ ; q ˙; 2 L2 MM dtdxdvdv d ; h 2 L1loc dtdxI L1 .1 C jvj2 /Mdv ; any joint limit points of the families gO ˙ , qO ˙; and h defined by (5.3), (5.6) and (5.29), respectively. 2 Then, one has h D j v C w jvj2 32 and Z Z 0 ˙; ˙2 q M dv d D L.h/ D h h0 h C h M dv d R3 S2 R3 S2 2 jvj D j L .v/ wL ; 2 (6.11)

189


where j and w are, respectively, the bulk velocity and temperature associated with the limiting fluctuation h, i.e., j is the electric current and w is the internal electric energy. Proof. We start from the decomposition Z

ˇ C ˇ2 ı ˇˇ C ˇˇ2 2 2 O ˇ ˇ h D h C gO jgO j gO jgO j M dv ; 4 R3

(6.12)

which follows from the decomposition (5.5) of fluctuations. In particular, integrating 2 (6.12) against vMdv and jvj3 1 Mdv yields Z

ˇ C ˇ2 ı ˇgO ˇ jgO j2 vM dv; O j D j C 4 R3 Z

jvj2 ˇ C ˇ2 ı 2 ˇ ˇ 1 M dv: gO jgO j w D wO C 4 R3 3 According to Lemma 5.14, we consider now weakly convergent subsequences h * h;

O hO * h;

in L1loc .dtdxI L1 ..1 C jvj2 /Mdv//, and j * j;

w * w;

jO * jO;

wO * w; O

in L1loc .dtdx/. Clearly, since ı D o.1/, we easily obtain, in view of the uniform L2loc .dtdxI L2 ..1 C jvj2 /Mdv// bound on gO ˙ provided by Lemma 5.12, passing to the limit in (6.12), that O h D h;

j D jO

and

w D w: O

Furthermore, using the relaxation estimate (5.11) from Lemma 5.10, it holds that ı C hO …hO D gO …gO C gO C …gO C D O.ı/L1 dt dxIL2 .Mdv/ ; loc O for ı vanishes asymptotically, and, therefore, whence hO D …h, 2 3 jvj : h D hO D j v C w 2 2 Next, it is readily seen that the elementary decompositions q q ˙ ˙ 2 ˙ L gO ; gO D Q gO ; gO Q G ; G ; 2 1 L gO ˙ ; gO D ˙ L hO C L gO C C gO ; 2ı 2

(6.13)

190


yield that q q ˙ 4ı ı C ˙ O L h D L gO C gO ˙ ıQ gO ; gO 2 Q G ; G C gO …gO C gO …gO D ı L CL Q gO ˙ ; gO Z 2 qO ˙; M dv d:

(6.14)

R3 S2

Therefore, passing to the limit ! 0 in (6.14), we find, in view of the control (5.11) from Lemma 5.10 and the fact that the linear and quadratic collision operators are continuous on L2 .Mdv/ (see (5.18)), that Z L .h/ D 2 q ˙; M dv d: R3 S2

Further employing the infinitesimal Maxwellian expression of h from (6.13), we arrive at Z 1 j L .v/ C wL jvj2 D 2 q ˙; M dv d; 2 R3 S2 which concludes the proof of the proposition.

6.3 Energy inequalities In view of the results from Sections 6.1 and 6.2, we are now able to establish the limiting energy inequalities for one species and for two species in the case of weak interactions only. The limiting energy inequality for strong interactions will require the results from Section 8.2 and, thus, will be treated later on in Section 8.3. Proposition 6.4. Let .f ; E ; B / be the sequence of renormalized solutions to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) considered in Theorem 4.5. In accordance with Lemmas 5.1, 5.2 and 5.3, denote by q 2 L2 MM dtdxdvdv d ; g 2 L1 dtI L2 .Mdxdv/ ; and E; B 2 L1 dtI L2 .dx/ ; any joint limit points of the families gO and qO defined by (5.3) and (5.6), E and B , respectively. Then, one has the energy inequality, for almost every t 0, 3 1 2 2 2 2 2 kkL2 C kukL2 C k kL2 C kEkL2 C kBkL2 .t/ x x x x x 2 2 Z t 5 C krx uk2L2 C krx k2L2 .s/ ds C in ; x x 2 0

6.3 Energy inequalities

191

where , u and are, respectively, the density, bulk velocity and temperature associated with the limiting fluctuation g, and the viscosity > 0 and thermal conductivity > 0 are defined by (2.29). Proof. First, by the estimate (5.7) from Lemma 5.3 and the weak sequential lower semi-continuity of convex functionals, we find that, for all t 0, Z Z Z 1 t q 2 MM dvdv d dxds 4 0 R3 R3 R3 S2 Z Z Z 1 t qO 2 MM dvdv d dxds lim inf !0 4 0 3 3 3 2 R R R S Z Z 1 t D .f / .s/ dxds; lim inf 4 !0 R3 0 which, when combined with Lemma 5.1, yields, passing to the limit in the entropy inequality (4.29), for almost every t 0, Z Z 2 1 1 2 g .t/M dxdv C jEj C jBj2 .t/ dx 2 R3 R3 2 R3 Z Z Z 1 t q 2 MM dvdv d dxds C in : C 4 0 R3 R3 R3 S2 2 Since, according to Lemma 5.10, the limiting fluctuation g D C u v C jvj2 32 is an infinitesimal Maxwellian, we easily compute that Z 3 g 2 M dv D 2 C juj2 C 2 ; 2 R3 which implies

3 1 2 2 2 2 2 kkL2 C kukL2 C k kL2 C kEkL2 C kBkL2 x x x x x 2 2 Z tZ Z 1 q 2 MM dvdv d dxds C in : C 4 0 R3 R3 R3 S2

(6.15)

There only remains to evaluate the contribution of the entropy dissipation in (6.15), which will result from a direct application of the following Bessel inequality (see [79, Chapter III, Section 4] for the basic principles of Bessel inequalities), established in [11, Lemma 4.7], and which follows from a projection on a suitable orthonormal set of L2 : ˇ2 ˇ2 ˇZ ˇZ ˇ ˇ ˇ 2 ˇˇ 8 ˇ ˇ Q q MM q Q MM dvdv d ˇˇ dvdv d ˇ C ˇ ˇ R3 R3 S2 5 R3 R3 S2 Z q 2 MM dvdv d; R3 R3 S2

where Q and Q are defined by (2.15).

(6.16)

192


For the sake of completeness and for later reference, we provide a short justification of (6.16) below. But prior to this, let us conclude the proof of the present proposition. To this end, we employ the identity (6.1) from Proposition 6.1 in combination with the relations (2.28), which we reproduce here for convenience: Z 2 ij Q kl M dv D ıi k ıj l C ıi l ıj k ıij ıkl ; 3 3 (6.17) ZR 5 Q M dv D ; ı i j ij 2 R3 to deduce from the inequality (6.16) that ˇ ˇ2 Z ˇ ˇ 2 2 t ˇ ˇ 2 ˇrx u C rx u .div u/ Idˇ C 10 jrx j q 2 MM dvdv d: 3 R3 R3 S2 Therefore, thanks to the solenoidal constraint on u established in (6.2), Z t 5 2 2 krx ukL2 C krx kL2 .s/ ds x x 2 0 Z tZ Z 1 q 2 MM dvdv d dxds: 4 0 R3 R3 R3 S2 Combining this with (6.15) concludes the proof of the proposition. Now, as announced above, we give a short proof of (6.16). To this end, following [11, Lemma 4.7], we recall that, for any traceless symmetric matrix A 2 R33 and any vector a 2 R3 , one computes straightforwardly, employing the identities (6.17) and the collisional symmetries, that Z 2 1 MM dvdv d A W Q C Q Q 0 Q 0 C a Q C Q Q 0 Q 0 3 3 2 16 R R S Z Z 1 1 ˝ Q M dv C .a ˝ a/ W ˝ Q M dv D .A ˝ A/ W 4 4 R3 R3 1 5 D A W A C a a: 2 8 Therefore, defining, for any q0 2 L2 .MM dvdv d /, the projection qN 0 D A0 W where

1Q 1Q C Q Q 0 Q0 C a0 C Q Q 0 Q 0 ; 4 4

Z 1 A0 D q0 Q C Q Q 0 Q 0 MM dvdv d; 2 R3 R3 S2 Z 2 a0 D q0 Q C Q Q 0 Q 0 MM dvdv d; 5 R3 R3 S2


193

we find that Z

1 5 q0 qN 0 MM dvdv d D A0 W A0 C a0 a0 2 8 R3 R3 S2 Z D qN 02 MM dvdv d: R3 R3 S2

Hence, we have the Bessel inequality 5 1 A0 W A0 C a0 a0 D 2 8

Z Z

R3 R3 S2

R3 R3 S2

qN 02 MM dvdv d (6.18) q02 MM dvdv d:

Therefore, setting q0 D q in (6.18), we find, exploiting the collisional symmetries of q, that Z Z 2 Q Q q MM dvdv d W q MM dvdv d R3 R3 S2 R3 R3 S2 Z Z 8 C q Q MM dvdv d q Q MM dvdv d 5 R3 R3 S2 R3 R3 S2 Z q 2 MM dvdv d; R3 R3 S2

which concludes the justification of (6.16). ˙ Proposition 6.5. Let f ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.6 for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded. In accordance with Lemmas 5.1, 5.2, 5.3 and 5.14, denote by h 2 L1loc dtdxI L1 1 C jvj2 Mdv ; g ˙ 2 L1 dtI L2 .Mdxdv/ ; and E; B 2 L1 dtI L2 .dx/ ; q ˙ ; q ˙; 2 L2 .MM dtdxdvdv d / any joint limit points of the families gO ˙ , h , qO ˙ and qO ˙; defined by (5.3), (5.29) and (5.6), E and B , respectively. Then, one has the energy inequality, for almost every t 0,

1 2 kuk2L2 C 5 k k2L2 C kEk2L2 C kBk2L2 .t/ x x x x 2 Z t 1 1 2 2 2 2 2 krx ukL2 C 5 krx kL2 C kj kL2 C C kwkL2 .s/ ds C in ; x x x x 2

0 where , u and are, respectively, the density, bulk velocity and temperature associated with the limiting fluctuation g, while j and w are, respectively, the electric

194


current and the internal electric energy associated with the limiting fluctuation h, and, finally, the viscosity > 0, thermal conductivity > 0, electric conductivity > 0 and energy conductivity > 0 are respectively defined by (2.61), (2.75) and (2.76). Proof. First, by the estimate (5.7) from Lemma 5.3 and the weak sequential lower semi-continuity of convex functionals, we find that, for all t 0, Z Z Z ˙ 2 1 t MM dvdv d dxds q 4 0 R3 R3 R3 S2 Z Z Z ˙ 2 1 t qO MM dvdv d dxds lim inf !0 4 0 R3 R3 R3 S2 Z Z 1 t D f˙ .s/ dxds; lim inf 4 !0 R3 0 and

1 2

Z tZ

Z

˙; 2 MM dvdv d dxds q 0 Z tZ Z ˙; 2 1 MM dvdv d dxds qO lim inf !0 2 0 R3 R3 R3 S2 Z Z ı2 t D fC ; f .s/ dxds; lim inf 4 !0 R3 0 R3

R3 R3 S2

which, when combined with Lemma 5.1, yields, passing to the limit in the entropy inequality (4.36), for almost every t 0, Z Z 2

2 1 1 g C C .g /2 .t/M dxdv C jEj C jBj2 .t/ dx 2 R3 R3 2 R3 Z tZ Z 2 1 2 q C C .q /2 C q C; C 4 0 R3 R3 R3 S2 2 MM dvdv d dxds C q ;C C in : Since, according to Lemmas 5.10 and 5.11 and Proposition 6.2, the limiting fluctuations 2 3 jvj ˙ g DCuvC 2 2 are infinitesimal Maxwellians which coincide, we easily compute that, in view of the strong Boussinesq relation C D 0 following from (6.8), Z

C 2 3 5 1 C .g /2 M dv D 2 C juj2 C 2 D juj2 C 2 ; g 3 2 R 2 2


195

which implies 5 1 1 2 2 2 2 kukL2 C k kL2 C kEkL2 C kBkL2 x x x x 2 2 2 Z tZ Z 2 C 2 1 (6.19) C .q /2 C q C; q C 4 0 R3 R3 R3 S2 2 C q ;C MM dvdv d dxds C in : There only remains to evaluate the contribution of the entropy dissipation in (6.19). To this end, applying the method of proof of Proposition 6.4, based on the Bessel inequality (6.16), with the constraints (6.7) and (6.8) from Proposition 6.2, note that Z t 5 2 2 krx ukL2 C krx kL2 .s/ ds x x 2 0 (6.20) Z tZ Z ˙ 2 1 MM dvdv d dxds: q 4 0 R3 R3 R3 S2 Next, the contributions of the mixed entropy dissipations q ˙; will be evaluated through a direct application of the following Bessel inequality: ˇZ ˇ2 ˇZ ˇ2 ˇ ˇ ˇ ˇ ˙; ˙; 2 ˇ ˇ ˇ 2 ˇ q vMM dvdv d ˇ C ˇ q jvj MM dvdv d ˇˇ R3 R3 S2 R3 R3 S2 Z ˙; 2 MM dvdv d: q R3 R3 S2

(6.21) For the sake of completeness, we provide a short justification of (6.21) below. But prior to this, let us conclude the proof of the present proposition. To this end, we employ the identity (6.11) from Proposition 6.3 in combination with the relations (2.75) and (2.76) to deduce from the inequality (6.21) that Z ˙; 2 2 1 MM dvdv d: q jj j2 C jwj2 (6.22)

R3 R3 S2 Combining this with (6.19) and (6.20) concludes the proof of the proposition. Now, as announced, we give a short proof of (6.21). To this end, for any vector A 2 R3 and any scalar a 2 R, one computes straightforwardly, employing Proposition A.1 and the collisional symmetries, that Z 2 A v v v 0 C v0 C a jvj2 jv j2 jv 0 j2 C jv0 j2 MM dvdv d R3 R3 S2 Z 2 D4 A v v 0 C a jvj2 jv 0 j2 MM dvdv d R3 R3 S2

16 8 D jAj2 C a2 :

196


Therefore, defining, for any q0 2 L2 .MM dvdv d /, the projection qN 0 D A0 v v v 0 C v0 C a0 jvj2 jv j2 jv 0 j2 C jv0 j2 ; where

Z A0 D q0 v v v 0 C v0 MM dvdv d; 3 3 2 8 R R S Z

q0 jvj2 jv j2 jv 0 j2 C jv0 j2 MM dvdv d; a0 D 16 R3 R3 S2

we find that Z Z 8 16 q0 qN0 MM dvdv d D jA0 j2 C a02 D qN 02 MM dvdv d: 3 3 2 3 3 2

R R S R R S Hence, we have the Bessel inequality 16 8 jA0 j2 C a02 D

Z Z

R3 R3 S2

R3 R3 S2

qN 02 MM dvdv d (6.23) q02 MM dvdv d:

Therefore, setting q0 D q ˙; in (6.23), we find, exploiting the collisional symmetries of q ˙; , that ˇ2 ˇZ Z 2 ˇ ˇ q ˙; vMM dvdv d ˇˇ C

q ˙; jvj2 MM dvdv d 2 ˇˇ R3 R3 S2 R3 R3 S2 Z ˙; 2 MM dvdv d; q R3 R3 S2

which concludes the justification of (6.21).

6.4 The limiting Maxwell equations Using the uniform L1 .dtI L2 .dx// bounds on the electromagnetic fields E and B , and the controls from Chapter 5 on the fluctuations, we can also take limits in the full Maxwell system for one species and for two species in the case of weak interactions only. Because of the scaling of the light speed, we obtain different kinds of limiting systems in the two regimes to be considered, but there is no particular difficulty here, for everything remains linear. As for the case of two species with strong interactions, we will not be able to pass to the limit in Maxwell’s equations. Indeed, Ampère’s equation is nonlinear in this setting, which is a major obstacle to the weak stability of the system. More comments on this issue are provided below.

6.4 The limiting Maxwell equations

197

In the regime leading to the incompressible quasi-static Navier–Stokes–Fourier– Maxwell–Poisson system, considered in Section 4.2, we start from Z 8 ˆ E rot B D g vM dv; @ ˆ t ˆ ˆ R3 ˆ ˆ < @ B C rot E D 0; t Z ˆ ˆ g M dv; div E D ˆ ˆ ˆ R3 ˆ : div B D 0: Then, the weak compactness of the fluctuations from Lemma 5.1, inherited from the scaled entropy inequality (4.29), allows us to consider converging subsequences E * E in L1 dtI L2 .dx/ ; B * B in L1 dtI L2 .dx/ ; g * g in L1loc dtdxI L1 1 C jvj2 M dv ; which easily leads to rot E D 0;

div E D ;

rot B D u;

div B D 0;

where and u respectively denote the density and bulk velocity associated to the limiting fluctuation g. Formally, this limit amounts to discarding the terms involving time derivatives in Maxwell’s equations, which accounts for the terminology of “quasi-static approximation” since temporal variations are neglected. Next, in the regime leading to the two-fluid incompressible Navier–Stokes– Fourier–Maxwell system with (solenoidal) Ohm’s law, considered in Section 4.3, we start from 8 Z Z C ı ˆ ˆ h vM dv; g g vM dv D @t E rot B D ˆ ˆ ˆ R3 R3 ˆ ˆ < @ B C rot E D 0; t Z C ˆ ˆ ˆ div E D ı g g M dv; ˆ ˆ ˆ R3 ˆ : div B D 0: We consider first the simpler case of weak interspecies collisions, i.e., the case ı D o.1/ and ı unbounded. The weak compactness of the fluctuations from Lemmas 5.1 and 5.14, inherited from the scaled entropy inequality (4.36), allows us to consider converging subsequences E * E in L1 dtI L2 .dx/ ; in L1 dtI L2 .dx/ ; B * B g˙ * g ˙ in L1loc dtdxI L1 1 C jvj2 M dv ; in L1loc dtdxI L1 1 C jvj2 M dv ; h * h

198


which easily leads to

8 @t E rot B ˆ ˆ ˆ < @ B C rot E t ˆ div E ˆ ˆ : div B

D j; D 0; D 0; D 0;

where j denotes the electric current, that is the bulk velocity associated to the limiting fluctuation h. Now, we see that in the case of strong interspecies collisions, i.e., ı D 1, Lemma 5.14 provides no longer enough compactness on h to take weak limits in Ampère’s equation. Indeed, in view of Lemma 5.13, it holds, at best, that the h ’s are uniformly bounded in L1loc .dtdxI L1 ..1 C jvj2 /Mdv//, but nothing prevents the fluctuations h from concentrating on small sets and, therefore, to converge towards a singular measure. Thus, in this asymptotic regime, Ampère’s equation will not be satisfied in the sense of distributions, but only in a dissipative sense, which will be encoded in the inequality defining the dissipative solutions obtained in Chapter 12 by means of a generalized relative entropy method. A closer inspection of Ampère’s equation in the limiting system (4.38) (or (2.80)) provides some insight on its lack of weak stability in the hydrodynamic limit. Indeed, even though Maxwell’s system in (4.38) is linear in the variables .E; B; n; j /, the energy inequality associated with (4.38) suggests that the right mathematical variables are rather .E; B; j nu; n; u/, which renders Ampère’s equation nonlinear. Nevertheless, the rest of Maxwell’s system remains linear and we can easily pass to the limit in Faraday’s equation and Gauss’ laws. Indeed, the weak compactness of the fluctuations from Lemma 5.1, inherited from the scaled entropy inequality (4.36), allows us to consider converging subsequences E * E in L1 dtI L2 .dx/ ; in L1 dtI L2 .dx/ ; B * B g˙ * g ˙ in L1loc dtdxI L1 1 C jvj2 M dv ; which easily leads to

8 ˆ < @t B C rot E D 0; div E D n; ˆ : div B D 0;

where n D C is the electric charge associated with the limiting fluctuations g ˙ , i.e., ˙ are the macroscopic densities of g ˙ .

Chapter 7

Strong compactness and hypoellipticity In Chapter 5, we have established uniform bounds and relaxation estimates on the fluctuations and collision integrands as consequences of the scaled relative entropy inequalities (4.29) and (4.36). This is sufficient to handle linear terms. Thus, in Chapter 6, we exploited these uniform estimates to derive limiting constraint equations and energy inequalities. In order to go any further in the rigorous derivation of the hydrodynamic limits under study, we need now to obtain precise strong compactness estimates on the fluctuations through a refined understanding of the Vlasov–Boltzmann equations from (4.28) and (4.35). More precisely, in the present chapter, we are going to introduce mathematical tools used to study the dependence in x and v of the families of fluctuations, and then deduce important strong compactness properties of these fluctuations. The first and simplest step, performed in Section 7.1 below, consists in understanding the dependence of fluctuations with respect to the velocity variable, which is essentially controlled by the relaxation mechanism. Since these estimates in v are based only on results from functional analysis and on the relative entropy and entropy dissipation bounds, they will hold similarly in both regimes (4.28) and (4.35). This first step is novel and differs considerably from previous works on hydrodynamic limits of Boltzmann equations with cutoff assumptions. Indeed, we establish below the strong compactness of the fluctuations in velocity, whereas former results only employed weak bounds in v, such as the equi-integrability in v from Lemma 5.12. This strong compactness is crucial in order to carry out the next stage of the proof in Section 7.2. Note that strong velocity compactness was also used in [4] to treat hydrodynamic limits of the Boltzmann equation without any cutoff assumptions. The approach therein relied heavily on the smoothing effect in v peculiar to long-range interactions, though. In fact, the methods developed in the present book can also be used to improve the results from [4] (see Part III). The second, more convoluted step, performed in Section 7.2, uses then the hypoellipticity in kinetic transport equations studied in [7] to transfer strong compactness from the velocity variable v to the space variable x. Some non-trivial technical care will be required in order to extend the results from [7], which mainly concern the stationary kinetic transport equation, to the non-stationary transport equation with a vanishing time derivative. Note that this second step also differs substantially from previous works on the subject, for these traditionally relied on classical velocity averaging lemmas to show some strong space compactness of the moments of the fluctuations (not the fluctuations themselves). It is to be emphasized that the compactness properties for the two-species regime (4.35) obtained in Section 7.2.3 below are substantially weaker than those corresponding to the one-species regime (4.28) and derived in Section 7.2.2. Essentially,

200

7 Strong compactness and hypoellipticity

the two-species regime considered here being quite singular, the corresponding fluctuations cannot be shown to enjoy as much equi-integrability as in the one-species regime, which will lead to significant difficulties in the remainder of our proofs. The results from the present chapter constitute a crucial and difficult step in the rigorous proofs of hydrodynamic convergence. They will allow us to obtain higherorder nonlinear constraint and evolution equations in the coming chapters. Finally, note that the results obtained here are only concerned with the compactness properties of fluctuations in x and v, but not in t. In fact, there may be oscillations in time and the temporal behavior of fluctuations will be analyzed later on in Chapter 10.

7.1 Compactness with respect to v We have already shown in Section 5.4 how the relaxation process towards statistical equilibrium provides improved integrability in v on the fluctuations. We show now how it further yields dissipative properties in the velocity variable. Loosely speaking, such a dissipation mechanism stems from the fact that the entropy dissipation controls the distance from the solutions to the set of statistical equilibria, which are in general smooth distributions in velocity. As we consider fluctuations around a global equilibrium, the linearized collision operator will play a fundamental role, just as in Section 5.3 on the relaxation. In order to get strong compactness results, we will further need to control the correctors coming from the nonlinear part of the collision operator. To this end, we recall now the important regularizing effects of the gain term of the Boltzmann collision operator. This property will be crucial in our proof of compactness. The results presented in Section 7.1.1 below concern general cross-sections satisfying some integrability assumptions. The properties from Section 7.1.2, however, only concern the Maxwellian collision kernel b 1. The corresponding results for general cross-sections will be discussed in the remaining parts of our work.

7.1.1 Compactness of the gain term In [53], Lions exhibited the compactifying and regularizing effects of the gain term of the Boltzmann collision operator. The essential result contained therein establishes the regularity of the gain term for a smooth and truncated collision kernel. The precise result from [53] which is of interest to us is recalled in the following theorem. Variants and refinements of this result were obtained in [13, 77]. In particular, a simple argument based on the Fourier transform, due to Bouchut and Desvillettes in [13], also provides a convenient compactness result. z Theorem 7.1 ([53]). Let b.z; / D b jzj; jzj 2 Cc1 ..0; 1/.0; // be a smooth compactly supported collision kernel. Then, there exists a finite C > 0 such that C Q .f; g/ C kf k kgk ; H 1 R3 L2 R3 L1 R3

7.1 Compactness with respect to v

201

for any f 2 L2 .R3 / and g 2 L1 .R3 /, and C Q .f; g/ C kf k kgk ; H 1 R3 L1 R3 L2 R3 for any f 2 L1 .R3 / and g 2 L2 .R3 /. Note that, in the statement of the above theorem, we have carefully avoided the endpoints on the domain of definition of the collision kernel in order to restrict the compact support of b.z; /. More precisely, Lions’ result ˇ z only ˇ considers smooth ˚ ˇ < 1 , for some kernels whose support is contained in < jzj < 1 ; ˇ jzj small > 0. This hypothesis is definitely not optimal, but at least some truncation is clearly required in order to obtain the optimal gain of regularity for QC .f; g/. For more general collision kernels, it is still possible to obtain some compactness of the gain operator by standard approximation procedures based on convolution inequalities for the gain term QC .f; g/. See for instance [3] for such general convolution inequalities. Here, we will merely use an elementary version of these inequalities which we presently recall for convenience. Thus, let 1 s p; q r 0 1 be such that 1C

1 1 1 1 D C C s p q r 0

and consider f 2 Lp .R3 /, g 2 Lq .R3 /, ' 2 Ls .R3 / and a general collision kernel b.z; / 2 Lr .R3 I L1 .S2 //. Then, employing the collision symmetries with Hölder’s and Young’s inequalities and using the change of variables v 7! V D v v , we find that ˇ ˇZ ˇ ˇ C ˇ ˇ ˇ 3 Q .f; g/.v/'.v/ dv ˇ R Z ˇ ˇ ˇfg ' 0 ˇ b.v v ; / dvdv d 3 3 2 ˇ ZR R S ˇ ˇ V jV j ˇˇ ˇ D ˇf .v/g.v V /' v 2 C 2 ˇ b.V; / d Vdvd R3 R3 S2 Z Z b.V; / d d V k'kLs0 kf .v/g.v V /kLsv R3 S2 Z b.V; /d k'kLs0 kf .v/g.v V /kLsv Lr 0 r 2 V S

LV

1 Z s D k'kLs0 b.V; /d jf .v/js jg.v V /js dv r0 r R3 S2 LVs LV Z k'kLs0 kf kLp kgkLq 2 b.V; /d r : Z

S

LV

202


Notice that the exact same reasoning can be applied to the loss operator Q .f; g/, so 0 that, considering the supremum over all ' 2 Ls R3 , we arrive at the estimate Z ˙ Q .f; g/ s kf k p kgk q b.z; /d (7.1) L L : L S2

Lrz

It turns out that it is possible to extend the above inequality to the full range of parameters 1 p; q; r; s 1 for the gain term QC .f; g/ only, provided we have a better control on the angular collision kernel. This is consistent with the fact that QC .f; g/ behaves nicely and better than Q .f; g/. Such results can be found in [3]. Thus, combining the regularizing properties from Theorem 7.1 with the convolution inequalities (7.1), we obtain the following convenient proposition. Proposition 7.2. Let b.z; / be a cross-section such that Z b.z; /d 2 L2 R3 ; M.z/˛ dz ; S2

for some given ˛ < 12 . Then, the bilinear operator L2 R3 ; Mdv L2 R3 ; Mdv ! L2 R3 ; M 1C2˛ dv QC .f; g/ .f; g/ 7! is locally compact. That is to say, it maps bounded subsets of L2 .Mdv/ L2 .Mdv/ into relatively compact subsets of L2loc .dv/. Proof. First, it is easy to check that p

0 p

0 1 1 .Mf /0 .Mg/0 M ˛ 2 D Mf M g M ˛ M2 p

0 p

0 ˛ C Mf Mg MM p

0 p

0 ˛ C M jf j M jgj .M.v v // 2 ;

for some C > 0. Hence, by virtue of the convolution inequality (7.1), we obtain C Q .f; g/ M ˛ 2 L .Mdv/ Z

0 p

0 p ˛ 2 dv d C M jf j M jgj b.v v ; /M.v v / 3 2 2 R S L .dv/ Z C kf kL2 .Mdv/ kgkL2 .Mdv/ b.z; / d ; S2

L2 M ˛ dz

(7.2) which establishes the boundedness of the quadratic operator.


203

Next, in order to show the local compactness of the operator, we consider any bounded sequences ffn gn2N ; fgn gn2N L2 .Mdv/. Then, defining QC .fn ; gn /, for any > 0, by simply replacing b.z; / by some smooth kernel b .z; / D z 2 Cc1 ..0; 1/ .0; // such that 0 b b and b jzj; jzj Z

S2

jb b j.z; /d

L2 M ˛ dz

< ;

we deduce, thanks to (7.2), that the terms QC .fn ; gn / can be uniformly approx2 1C2˛ imated by the terms QC dv/. Since, by Theorem 7.1, the .fn ; gn / in L .M C Q .Mfn ; Mgn /’s are relatively compact in L2loc .dv/, we conclude that the original sequence fQC .Mfn ; Mgn /gn2N is relatively compact in L2loc .dv/, which concludes the justification of the proposition.

7.1.2 Relative entropy, entropy dissipation and strong compactness Combining the uniform controls from the relative entropy and entropy dissipation with the compactness of the gain term presented in the previous section, we establish now the following result, valid for the Maxwellian collision kernel b 1. Lemma 7.3. Let f .t; x; v/ be a family of measurable, almost everywhere nonnegative distribution functions such that, for all t 0, Z Z 1 t 1 H f D .f / .s/ dxds C in : .t/ C 2 4 0 R3 Then, as ! 0, any subsequence of renormalized fluctuations gO is locally relatively compact in v in L2 .dtdxdv/ in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup kgO .t; x; v C h/ gO .t; x; v/kL2 .K;dt dxdv/ < : >0

Proof. Loosely speaking, the present proof can be summarized in three main steps, each corresponding to a decomposition of gO . First, we will show how to control the very large values of gO , i.e., values larger than 1 , with the entropy bound. This is a rather standard and simple estimate. Second, we derive the strong compactness in velocity of gO from the entropy dissipation bound and the compactness of the gain term (see Proposition making sure that we remain away from vacuum, i.e., away R 7.2) from the values 1 C 2 gO M dv 0, for our estimates degenerate in this case. Finally, we deduce the strong compactness near vacuum arguing that the vacuum state gO 2 is actually smooth since it is constant. Control of very large values. Using the entropy inequality, we first introduce some microscopic truncation of large values. For any fixed small 0 < < 12 and any

204


cutoff function .r/ 2 Cc1 .R/ such that 1fjrj1g .r/ 1fjrj2g , we have, since 2 gO 2 4G as soon as gO 1, jgO .1 . gO //j2 gO 2 1fgO 1 >2g

4 2 G 1nG > 1 o 42 2 G log G 1nG > 1 o ; 2 jlog 2 j 42 so that, by the relative entropy bound, gO .1 . gO // D O

!

1 1

j log j 2

(7.3)

L1 dt IL2 .Mdxdv/

as ! 0, uniformly in . Away from vacuum. We use now the Hilbert decomposition (see Proposition 5.4) for the Maxwellian cross-section b 1:

LgO D gO KgO ; where K is a compact integral operator on L2 .Mdv/. Then, from the identity p 2 p LgO D Q .gO ; gO / Q G ; G ; 2 we deduce that gO D LgO C KgO Z Z D QC .gO ; gO / gO gO M dv d qO M dv d C KgO ; 2 2 R3 S2 R3 S2 or, equivalently, Z Z gO 1 C gO M dv D qO M dv d C KgO C QC .gO ; gO / : 2 R3 2 R3 S2 (7.4) We are now going to control each term in the right-hand side above separately. The first term is easily estimated employing the uniform L2 -estimate from Lemma 5.3. It yields that Z qO M dv d D O ./L2 .Mdt dxdv/ : (7.5) R3 S2

The second term KgO satisfies the bound kKgO kL2 .Mdv/ C kgO kL2 .Mdv/ ;


so that, in view of Lemma 5.2 and of the compactness of the operator K, ; KgO D O.1/ L1 dt IL2 dxIC L2 .Mdv/

205

(7.6)

where we have used the notation C L2 .Mdv/ to indicate that it is relatively compact with respect to the velocity variable in L1 .dtI L2 .Mdxdv//. Similarly, the third term 2 QC .gO ; gO / satisfies, by virtue of Proposition 7.2, the control C Q .gO ; gO / D O./L1 dt IL1 dxIC L2 .dv/ ; 2 loc

(7.7)

where, again, we have used the notation C L2loc .dv/ to indicate that it is relatively compact with respect to the velocity variable in L1 .dtI L1 .dxI L2loc .dv///. On the whole, incorporating the controls (7.5), (7.6) and (7.7) into the decomposition (7.4), we have established that Z gO 1 C gO M dv D O ./L2 .Mdt dxdv/ 2 R3 (7.8) C O.1/L1 dt IL2 dxIC L2 .Mdv/ C O./L1 dt IL1 dxIC L2

:

loc .dv/

Next, since the left-hand side of the R above decomposition degenerates close to vacuum, i.e., whenever the density 2 R3 gO M dv is close to 1, we introduce a macroscopic truncation ;r .t; x/ D 1f1C R M.v/gO .t;x;v/dvr g ; 2

for some small r > 0, thus excluding Rthe domain where this degeneracy is present. It then follows, dividing (7.8) by 1 C 2 R3 gO M dv , that ;r gO D O r L2 .Mdt dxdv/ 1 CO (7.9) r L1 dt IL2 dxIC L2 .Mdv/ : CO r L1 dt IL1 dxIC L2loc .dv/ Next, for any small h 2 R3 and any compact subset K Œ0; 1/ R3 R3 , it holds that Z ;r j. gO /gO .t; x; v C h/ . gO /gO .t; x; v/j2 dtdxdv K Z D2 ;r . gO /gO .t; x; v/ . gO /gO .t; x; v/

K . gO /gO .t; x; v C h/ dtdxdv Z ;r j. gO /gO .t; x; v/ .gO .t; x; v C h/ gO .t; x; v//j dtdxdv: C K

206


Therefore, since

k. gO /gO kL2

loc .dt dxdv/

C;

2 ;

we conclude from (7.9) that, for any fixed 0 < ; r < 1, Z lim sup lim sup ;r j. gO /gO .t; x; v C h/ . gO /gO .t; x; v/j2 dtdxdv D 0; k. gO /gO kL1 .dt dxdv/

jhj!0

!0

K

the expected relative compactness statement away from vacuum on ;r . gO /gO . Consequently, combining this result with the control (7.3) on the very large values of gO yields that, for any given small r > 0, ;r gO is relatively compact with respect to the velocity variable in L2loc .dtdxdv/:

(7.10)

Near vacuum. It only remains then to get a compactness estimate near vacuum on .1 ;r /gO . To this end, we simply decompose, for any given small > 0, . gO /gO D . gO /

2 2 2 1 C gO C .1 . gO // ; 2

(7.11)

and we control each term in the right-hand side above individually. Thus, noticing that, for any 0 < r 12 , R M gO dv 1 n 1 R o .1 ;r / D 1 2 < M gO dv 2.1 r/ 1r kgO kL2 .Mdv/ D O.1/L1 dt IL2 .dx/ ; and that, on the support of 1 ;r , Z Z ˇ 1 ˇˇ2 ˇ M 1 C gO dv 1 C ˇ. gO / 1 C gO ˇ M dv 2 2

R3 R3 r 1 ; r 1C DO

L1 .dt dx/ we obtain concerning the first term in the right-hand side of (7.11) that r 2 r .1 ;r /. gO / 1 C gO D O : 2

L1 dt IL2 .Mdxdv/ Then, the second term is easily handled through the estimate 1 1 .1 . gO // 1fjgO j>1g jgO j ;

(7.12)

7.2 Compactness with respect to x

207

whereby 2 D O . /L1 dt IL2 .Mdxdv/ : (7.13) The remaining term in the right-hand side of (7.11) is constant, in particular it is smooth, and so there is no need to further control it. Hence, on the whole, incorporating (7.12) and (7.13) into (7.11), we find r r .1 ;r /. gO /gO D O C O . /L1 dt IL2 .Mdxdv/

L1 dt IL2 .Mdxdv/ .1 ;r / .1 . gO //

2 .1 ;r /: We therefore conclude, for any small h 2 R3 and any compact subset K Œ0; 1/ R3 R3 , that Z lim sup lim sup .1 ;r / j. gO /gO .t; x; v C h/ . gO /gO .t; x; v/j2 dtdxdv jhj!0

O

!0

r

K

C O 2 :

(7.14) Conclusion of proof. On the whole, combining the above estimate (7.14) near vacuum with the control (7.3) on the very large values of gO and the compactness statement (7.10) away from vacuum, we finally arrive at the control, for any compact subset K Œ0; 1/ R3 R3 , Z lim sup lim sup jgO .t; x; v C h/ gO .t; x; v/j2 dtdxdv jhj!0

O

!0

r

K

C O 2 C O

1 ; j log j

which, by the arbitrary smallness of r > 0 and > 0, clearly implies that gO is locally relatively compact with respect to the velocity variable in L2 .dtdxdv/ and thus concludes the proof of the lemma.

7.2 Compactness with respect to x In order to get a refined description of the dependence of the fluctuations g and g˙ with respect to x, we will use the compactness properties of the free transport operator v rx . More precisely, there are two types of mechanisms at play here: the transfer of compactness, which expresses the fact that the free transport mixes the spatial and velocity variables and is a consequence of hypoellipticity;

208


the averaging lemma, which predicts some regularizing effect for the averages with respect to v, due to the fact that the symbol of the free transport is elliptic on a large microlocal subset. Of course both mechanisms require that we have a good control on the advection terms v rx gO and v rx gO ˙ or some similar quantity (since the square root renormalization is singular at the origin, and thus is not admissible; see proofs of Lemmas 7.8 and 7.10 below). In particular, we see at this point that the situation is quite different in the one-species scaling leading to the incompressible quasistatic Navier–Stokes–Fourier–Maxwell–Poisson system, and in the two-species scaling leading to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with (solenoidal) Ohm’s law. In the first case, at the formal level, it is natural to expect from (4.30) that, up to some suitable renormalization, the advection term v rx g is uniformly bounded. However, for the multi-species model, we see that (4.37) provides, at least formally, some uniform control on the advection terms v rx g˙ , and therefore some strong ˙ compactness on the hydrodynamic variables ˙ , u˙ and , but it does not provide any information on v rx ı gC g which controls the electrodynamic variables j and w . As for the limiting systems (4.38) and (4.39), we therefore do not expect the electromagnetic terms j ^ B from the Lorentz force to be weakly stable (unfortunately, compensated compactness methods also fail here; see [6] for some details on this issue). In this case, we will use, later on in Chapter 12, some weak-strong stability principle instead of a priori estimates. In other words, the dependence with respect to x is partially understood a posteriori, by comparison with the solutions to the limiting systems.

7.2.1 Hypoellipticity and the transfer of compactness We first explain our global strategy, presenting the main abstract results we will use on the free transport operator. As mentioned in the introduction of the present chapter, the key idea here is to transfer the compactness with respect to v inherited from the structure of the collision operator (see Lemma 7.3) onto the spatial variable x. To this end, we need the following result. Theorem 7.4 ([7]). Let the bounded family of functions f .t; x; v/g2ƒ Lp Rt R3x R3v ; for some 1 < p < 1, be locally relatively compact in v and such that ˇ

˛

.@t C v rx / D .1 t;x / 2 .1 v / 2 S ; for all 2 ƒ and for some bounded family

fS .t; x; v/g2ƒ Lp Rt R3x R3v ;

where ˛ 0 and 0 ˇ < 1.


209

Then, f .t; x; v/g2ƒ is locally relatively compact in Lp .Rt R3x R3v / (in all variables). Remark. A closer inspection of the proof of the above theorem in [7] reveals that one actually has the following estimate, for any compact subset K R3 : lim sup

sup

!0 2ƒ jkjCjhjCjlj0 Lp Rt R3x R3v ; for some 1 < p < 1, be locally relatively compact in v and such that ˇ

˛

.@t C v rx / D .1 x / 2 .1 v / 2 S ; for all > 0 and for some bounded family

fS .t; x; v/g>0 Lp Rt R3x R3v ;

where ˛ 0 and 0 ˇ < 1. Then, f .t; x; v/g>0 is locally relatively compact in Lp .Rt R3x R3v / in x and v (but not necessarily in t). Proof. This result is directly deduced from Theorem 7.4 up to a change of variable in time. We will therefore loose the compactness with respect to time. More precisely, we define 1 Q .t; x; v/ D p .t; x; v/; SQ .t; x; v/ D p S .t; x; v/; 1

so that

ˇ

.@t C v rx / Q D .1 x / 2 .1 v / 2 SQ ; ˛

and Q and SQ are uniformly bounded in Lp .Rt R3x R3v /. We apply now Theorem 7.4 to the above transport equation to obtain that f Q g>0 is locally relatively compact in Lp .Rt R3x R3v /. More precisely, we deduce from (7.15), that D 0; Q .t C k; x C h; v C l/ Q .t; x; v/ lim sup sup Lp R R3 K !0 >0 jkjCjhjCjlj0 jhjCjlj0 jhjCjlj0 jkjCjhjCjlj0 L1 Rt R3x I Lr R3v ; for some 1 < r < 1, be locally relatively compact in v and such that ˇ

˛

.@t C v rx / D .1 x / 2 .1 v / 2 S ; for all > 0 and for some bounded family fS .t; x; v/g>0 L1 Rt R3x I Lr .R3v / ; where ˛ 0 and 0 ˇ < 1. We further assume that, for any compact set K R3 R3 , Z .t; x; v/ dxdv is equi-integrable (in t). K

>0

Then, f .t; x; v/g>0 is equi-integrable (in all variables) and locally relatively compact in L1 .Rt R3x R3v / in x and v (but not necessarily in t). Moreover, if the ’s are signed (in the sense that the functions may assume both positive and negative values), the conclusion still holds true, i.e., f .t; x; v/g>0 is locally relatively compact in L1 .Rt R3x R3v / in x and v (but not necessarily in t), provided f .t; x; v/g>0 is equi-integrable (in all variables) a priori. Proof. When the ’s are signed and a priori equi-integrable (in all variables), this result is deduced from Theorem 7.5 (and its proof) utilizing the strategy of proof of Lemma 7.6, that is, by dilation of the time variable. To this end, we define Q .t; x; v/ D .t; x; v/; SQ .t; x; v/ D S .t; x; v/; so that

ˇ

.@t C v rx / Q D .1 x / 2 .1 v / 2 SQ ; ˛

and Q and SQ are uniformly bounded in L1 .Rt R3x I Lr .R3v //.

212


We apply now Theorem 7.5 to the above transport equation to deduce that f Q g>0 is locally relatively compact in L1 .Rt R3x R3v /. In fact, a closer inspection of the proof of this theorem in [7] reveals that, by possibly localizing without loss of generality the above functions in v only, one has the following global estimate: lim sup

sup

!0 >0 jkjCjhjCjlj0 jhjCjlj 1, we have that k .t; x C h; v C l/ .t; x; v/kL1 .K/ Z 1 D jf.t; x; v/ 2 K W j .t; x C h; v C l/ .t; x; v/j > gj d

Z

0 R 1 R

jf.t; x; v/ 2 K W j .t; x C h; v C l/ .t; x; v/j > gj d C

jKj R

C . .t; x C h; v C l/ .t; x; v// 1fj .t;xCh;vCl/ .t;x;v/j>Rg L1 .K/ jKj 2 log R k .t; x C h; v C l/ .t; x; v/kL1;1 .K/ C R C . .t; x C h; v C l/ .t; x; v// 1fj .t;xCh;vCl/ .t;x;v/j>Rg L1 .K/ : Hence, we deduce, provided the ’s are equi-integrable in all variables and by the arbitrariness of R > 1, that lim sup

sup

!0 >0 jhjCjlj0 when it is not already known a priori and when each is non-negative. However, the preceding strategy based on time-dilations to deduce results from Theorem 7.5 cannot be repeated here, for the notion of equi-integrability does unfortunately not behave suitably under partial dilations. Instead, the proof of Theorem 2.4 from [7] has to be adapted to treat the present setting, which is rather involved. Therefore, in order to provide a self-contained justification based on [7] and for the sake of clarity, we have moved the remainder of the proof of the present lemma to Appendix C.


213

7.2.2 Compactness of fluctuations for one species The next step consists in combining the velocity compactness result from Lemma 7.3 with the hypoelliptic transfer of compactness contained in Lemma 7.7 to infer the compactness in x and v (but not in t) of the fluctuations gO and gO ˙ . To this end, we will need to consider the action of the transport operator .@t C v rx / on the fluctuations gO , gO ˙ (to control oscillations) and their square gO 2 , gO ˙2 (to control concentrations), or truncated versions of these fluctuations. Again, note that this strategy differs from the methods developed in previous works on hydrodynamic limits, since we do not use classical averaging lemma. Indeed, we prove below that the fluctuations themselves, and not only their moments with respect to v, are strongly compact in x and v. Let us first focus on the regime considered in Theorem 4.5 (with one species) leading to the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (4.31). In this case, we have the following equi-integrability property. Lemma 7.8. Let .f ; E ; B / be the sequence of renormalized solutions to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) considered in Theorem 4.5. Then, as ! 0, any subsequence of renormalized fluctuations gO is locally relatively compact in .x; v/ in L2 .dtdxdv/ in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup kgO .t; x C h; v C l/ gO .t; x; v/kL2 .K;dt dxdv/ < : >0

In particular, the family jgO j2 is equi-integrable (in all variables t, x and v). Proof. The proof of this lemma proceeds in two main steps. The first establishes the compactness of gO in x and v in L1loc , while the second shows the equi-integrability of gO 2 in all variables. The combination of these two steps will eventually allow us to conclude the proof. An admissible renormalization ˇ1 .G /. We consider first the admissible square root renormalization p z C a 1 ; ˇ1 .z/ D for some given 1 < a < 4. This renormalization is introduced to circumvent the fact that the natural renormalization p z1 2 corresponding to gO is not admissible for Vlasov–Boltzmann equations, for it is singular at z D 0, i.e., p 0 z1 2 !1 as z ! 0.

214


In fact, we have already used a similar strategy in the proof of Proposition 6.1, where we showed, as a consequence of the entropy and entropy dissipation bounds, that (see (6.5)) p

a G C a 1 gO D O 2 1 1 ; L .dt dxdv/ p a G C a 1 2 ; gO D O a1 L1 .dt dxdv/ C O 2 1 L .dt IL2 .Mdxdv// p a G C a 1 ; 2 gO D O a1 L1 .dt dxdv/ C O 2 2 Lloc dt dxIL2 1Cjvj2 Mdv (7.16) so that strong compactness properties of 2

p

G C a 1

p

in Lloc .dtdxdv/, for any given 1 p 2, will entail similar properties on gO in the same space and vice versa. Compactness of ˇ1 .G / in v. In particular, since, in view of Lemma 7.3, the renormalized fluctuations gO are locally relatively compact in v in L2loc .dtdxdv/, the same holds true for any subsequence of p

G C a 1 ;

in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup kˇ1 .G / .t; x; v C h/ ˇ1 .G / .t; x; v/kL2 .K;dt dxdv/ < :

(7.17)

>0

Action of the transport operator on ˇ1 .G /. Thus, using ˇ1 .z/ to renormalize the Vlasov–Boltzmann equation in (4.28) and decomposing the collision integrands according to (5.8), we find that (see (6.3)) p G C a 1 G E v p .@t C v rx C .E C v ^ B / rv / 2 G C a p Z Z p 2 G p D p G q O M dv d C qO 2 M dv d: 2 G C a R3 S2 8 G C a R3 S2 (7.18)


215

It follows, employing the uniform bounds gO 2 L1 .dtI L2 .Mdxdv// and qO 2 L2 .MM dtdxdvdv d / from Lemmas 5.2 and 5.3, respectively, that (see (6.4)) p G C a 1 .@t C v rx / p p G G C a 1 D p r E v 1 C .E C v ^ B / g O v 2 2 G C a p p Z Z G G qO M dv d C p gO qO M dv d C p 2 G C a R3 S2 4 G C a R3 S2 Z 2 C p qO 2 M dv d 8 G C a R3 S2 D O.1/L1 .dt dxdv/ C O./L1 dt dxIW 1;1 .dv/ : loc

loc

loc

(7.19) Compactness of ˇ1 .G / in .x; v/. On the whole, we have established the compactness in velocity of ˇ1 .G / in (7.17) and a bound on the transport operator acting on ˇ1 .G / in (7.19). Therefore, a direct application of Lemma 7.7 yields that p G C a 1 ˇ1 .G / D is locally relatively compact in .x; v/ in L1loc .dtdxdv/. Combining this result with (7.16), implies that the renormalized fluctuations gO are relatively compact in .x; v/ in L1loc .dtdxdv/ as well, in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup kgO .t; x C h; v C l/ gO .t; x; v/kL1 .K;dt dxdv/ < :

(7.20)

>0

An admissible renormalization ˇ2 .G /. In order to improve this local strong compactness in x and v from L1loc to L2loc , we only have to show now that gO 2 is locally equi-integrable in all variables, which will also be seen as a consequence of Lemma 7.7. To this end, we consider now the admissible renormalization p ˇ2 .z/ D

z C a 1

2

p

z C a 1 ;

where .z/ 2 C 1 .R/ is a cutoff satisfying 1Œ1;1 .z/ 1Œ2;2 , for some given 1 < a < 4 (in fact, we will further restrict the range of a so that necessarily a D 2 below) and any small enough > 0. As before, this renormalization is introduced to

216


circumvent the fact that the natural renormalization p 2 z1 2 corresponding to gO 2 is not admissible for Vlasov–Boltzmann equations, for it is singular at z D 0, i.e., " p 2 #0 z 1 ! 1 2 as z ! 0, and its growth at infinity is not admissible, i.e., p 2 z1 2 z as z ! 1. Next, it is readily seen that (7.16) implies that p 2 p G C a 1 G C a 1 2 gO D 2gO 2 gO 2 p 2 G C a 1 C 2 gO

a D O a1 C 2

(7.21) ;

1 L1 dt IL1 loc dxIL .Mdv/

so that the equi-integrability of p 2 p

G C a 1

G C a 1 ˇ2 .G / D p will entail the equi-integrability of gO 2 G C a 1 , which, when combined with the following control on the very large values of fluctuations (see (5.28)): p

.1 /

G C a 1 jgO j2 1f.pG Ca 1/>1g jgO j2 (7.22) 1 DO ; jlog j L1 dt IL1 .Mdxdv/ for any small enough > 0, will eventually imply the equi-integrability of gO 2 . Compactness of ˇ2 .G / in v. Furthermore, note that the velocity compactness stated in (7.17) implies a corresponding property for ˇ2 .G /. Indeed, writing ˇ2 .z/ D ˇ12 .z/ . ˇ1 .z// and noticing that, for any z1 ; z2 2 R and ˛ 2 R, ˇ ˇ 2 ˇz .˛z1 / z 2 .˛z2 /ˇ D j.z1 z2 /z1 .˛z1 / C z2 .z1 .˛z1 / z2 .˛z2 //j 1 2 C jz1 z2 j .jz1 j C jz2 j/ ; (7.23)

217


we deduce, for any h 2 R3 , that kˇ2 .G / .t; x; v C h/ ˇ2 .G / .t; x; v/kL1

loc .dt dxdv/

C kˇ1 .G /kL2

loc .dt dxdv/

kˇ1 .G / .t; x; v C h/ ˇ1 .G / .t; x; v/kL2

loc .dt dxdv/

:

It then follows from (7.17) that any subsequence of ˇ2 .G / is locally relatively compact in v in L1 .dtdxdv/ in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup kˇ2 .G /.t; x; v C h/ ˇ2 .G /.t; x; v/kL1 .K;dt dxdv/ < :

(7.24)

>0

However, in order to use the above velocity compactness of ˇ2 .G / in Lemma 7.7, we still need to show that ˇ2 .G / enjoys an improved integrability with respect to the velocity variable, namely that ˇ2 .G / is locally bounded in L1 .dtdxI Lr .dx//, for some r > 1. To this end, we introduce the decomposition p

1 G C a 1 p ˇ2 .G / D …gO G C a 1

2 p p

G C a 1 1 G C a 1 p G C a 1 C gO

2 p

p a G C 1 1 G C a 1 :

C .gO …gO / 2 Since …gO belongs to L1 .dtI L2 .dxI Lp .Mdv///, for any 1 p < 1, we therefore get, for any 1 r < 2, ˇp ˇ C ˇˇ G C a 1 1 ˇˇ C ˇ2 .G / O.1/L1 dt dxILr .Mdv/ C gO ˇ C jgO …gO j : loc

ˇ 2

Then, by (7.16) and the relaxation estimate (5.11), we obtain, provided 2 a < 4, ˇ2 .G / D O.1/ r L1 loc dt dxIL .Mdv/

CO DO

a2 2

!

CO

2 L1 loc dt dxIL .Mdv/

1 ;

L1loc dt dxILr .Mdv/

1

L1loc dt dxIL2 .Mdv/

for any 1 r < 2. Finally, combining the preceding estimate with the compactness estimate (7.24), we deduce, for any 1 r < 2, that ˇ2 .G / is locally relatively compact in v in L1 .dtdxI Lr .dv// in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup k.ˇ2 .G /.t; x; v C h/ ˇ2 .G /.t; x; v// 1K .t; x; v/kL1 .dt dxILr .dv// < : >0

(7.25)

218


Action of the transport operator on ˇ2 .G /. Thus, using ˇ2 .z/ to renormalize the Vlasov–Boltzmann equation in (4.28), decomposing the collision integrands according to (5.8) and writing for convenience .z/ D z 2. z/ C z 0 . z/ ; so that ˇ20 .z/ D ˇ10 .z/ .ˇ1 .z//, we have now that (note that this renormalization procedure amounts to multiplying (7.18) by .ˇ1 .G //) G .ˇ1 .G // .@t C v rx C .E C v ^ B / rv /ˇ2 .G / E v p 2 G C a p Z p G .ˇ1 .G // p D G qO M dv d 2 G C a R3 S2 Z 2 .ˇ1 .G // C p qO 2 M dv d: 8 G C a R3 S2 It follows, employing the uniform bounds gO 2 L1 .dtI L2 .Mdxdv// and qO 2 L2 .MM dtdxdvdv d / from Lemmas 5.2 and 5.3, respectively, and the direct estimates j .ˇ1 .G //j C jˇ1 .G /j D O .1/L2 .dt dxdv/ ; loc 1 .ˇ1 .G // D O ;

L1 .dt dxdv/ C 1 ; jˇ1 .G /j D O jˇ2 .G /j

L2loc .dt dxdv/ that, provided 1 < a 2, .@t C v rx /ˇ2 .G / p G .ˇ1 .G // p E v 1 C D gO 2 2 G C a rv Œ.E C v ^ B / ˇ2 .G / p Z G .ˇ1 .G // p C qO M dv d 2 G C a R3 S2 p Z G .ˇ1 .G // p gO qO M dv d C 4 G C a R3 S2 Z 2 .ˇ1 .G // qO 2 M dv d C p 8 G C a R3 S2 1 DO : 1;1

L1loc dt dxIWloc .dv/

(7.26)

Equi-integrability of ˇ2 .G / in .t; x; v/. On the whole, we have established the compactness in velocity of ˇ2 .G / in (7.25) and a bound on the transport operator


219

acting on ˇ2 .G / in (7.26). Therefore, further noticing that ˇ2 .G / is non-negative and, recalling gO 2 2 L1 .dtI L1 .Mdxdv// and the error estimate (7.21), that the R family K ˇ2 .G /dxdv is equi-integrable, for any compact set K R3 R3 , a direct application of Lemma 7.7 yields that p ˇ2 .G / D

G C a 1

2 p

G C a 1

is equi-integrable in all variables .t; x; v/. Combining this result with (7.21) and (7.22), implies that the renormalized fluctuations gO 2 are equi-integrable in all variables .t; x; v/ as well. Finally, further combining the equi-integrability of gO 2 with the local strong compactness estimate (7.20), we deduce that the renormalized fluctuations gO are relatively compact in .x; v/ in L2loc .dtdxdv/, in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup kgO .t; x C h; v C l/ gO .t; x; v/kL2 .K;dt dxdv/ < ; >0


Immediate consequences of the preceding strong compactness lemma are: the relative compactness in .x; v/ in L1 .dtdxdv/ of any subsequence of renormalized fluctuations gO 2 in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup gO 2 .t; x C h; v C l/ gO 2 .t; x; v/L1 .K;dt dxdv/ < : (7.27) >0

the nonlinear weak compactness property, for any p < 2, .1 C jvjp / gO 2 is weakly relatively compact in L1loc .dtdxI L1 .Mdv//; (7.28) which follows from the Dunford–Pettis compactness criterion (see [68]) by deducing the equi-integrability of gO 2 from Lemma 7.8 and the tightness of .1 C jvjp / gO 2 from Lemma 5.12. R the strong spatial compactness of the moments R3 gO '.v/M dv in L2loc .dtdx/, for any '.v/ 2 L2 ..1 C jvj2 /1 Mdv/, in particular lim sup kO .t; x C h/ O .t; x/kL2

jhj!0 >0

loc .dt dx/

D 0;

lim sup kuO .t; x C h/ uO .t; x/kL2 .dt dx/ D 0; loc lim sup O .t; x C h/ O .t; x/ 2 D 0:

jhj!0 >0

jhj!0 >0

Lloc .dt dx/

(7.29)

220


The next lemma is also a direct consequence of the strong compactness properties from the preceding lemma and concerns a refinement of the relaxation estimate (5.11) to L2loc .dtdxdv/. Lemma 7.9. Let .f ; E ; B / be the sequence of renormalized solutions to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) considered in Theorem 4.5. Then, as ! 0, any subsequence of renormalized fluctuations gO satisfies the relaxation estimate gO …gO ! 0 in L2loc dtdxI L2 .Mdv/ : Proof. On the one hand, we already know from Lemma 5.10 that gO …gO D O./L1 dt dxIL2 .Mdv/ :

(7.30)

loc

On the other hand, the uniform integrability in all variables of jgO j2 from Lemma 7.8 and the tightness in v of jgO j2 M implied by Lemma 5.12 show that Z Z 2 is uniformly integrable in t and x. j…gO j M dv C jgO j2 M dv R3

R3

Therefore, we deduce that kgO …gO k2L2 .Mdv/

is uniformly integrable in t and x.

(7.31)

Then, decomposing, for any large > 0, gO …gO D .gO …gO / 1nkgO

O kL2 .M dv/ …g

C .gO …gO / 1nkgO

o

O kL2 .M dv/ > …g

o;

we find that kgO …gO kL2 dt dxIL2 .Mdv/

p

loc

1

kgO …gO k 2 1 Lloc dt dxIL2 .Mdv/ o n C .gO …gO / 1 kgO …gO k >

L2 .M dv/

;

:


whence, by virtue of (7.30), lim sup kgO …gO kL2 dt dxIL2 .Mdv/ loc !0 n sup .gO …gO / 1 kgO …gO k >0

L2 .M dv/ >

o



221

Finally, thanks to the uniform integrability (7.31) and the arbitrariness of , we infer that D 0; lim kgO …gO k 2 L2 loc dt dxIL .Mdv/

!0


7.2.3 Compactness of fluctuations for two species We move on now to the study of strong compactness properties of the fluctuations considered in Theorems 4.6 and 4.7 leading to the two-fluid incompressible Navier– Stokes–Fourier–Maxwell systems with Ohm’s laws (4.38) and (4.39). Unlike the estimates from Chapter 5 infered from entropy and entropy dissipation bounds, here, we cannot deduce results for the two-species case from results for the one-species case. In fact, the regimes considered in Theorems 4.6 and 4.7 are much more singular than the regime studied in Theorem 4.5 and, as a result, the compactness properties asserted in Lemma 7.8 may not hold in the two-species case. It is to be emphasized that this lack of compactness is one the main drawbacks and difficulties preventing the improvement of Theorems 4.6 and 4.7 to a weak compactness result similar to Theorem 4.5. Recall, however, that such an improvement is not to be expected so readily since the limiting systems (4.38) and (4.39) are not stable under weak convergence in the energy space and, in particular, are not known to have global weak solutions (see corresponding discussion in Section 3.2). Thus, in the two-species case, we only have the following weaker strong compactness result. Lemma 7.10. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorems 4.6 and 4.7. Then, as ! 0, any subsequence of renormalized fluctuations gO ˙ is locally relatively compact in .x; v/ in Lp .dtdxdv/, for any 1 p < 2, in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup gO ˙ .t; x C h; v C l/ gO ˙ .t; x; v/Lp .K;dt dxdv/ < : >0

Furthermore, for any > 0, the families ˇ ˙ ˇ2 ˇgO ˇ 1n

ˇ o ˇ ˇ ˇ ıˇgO ˙ ˇ1

are equi-integrable (in all variables t, x and v).

222


Remark. We do not know whether the families jgO ˙ j2 are equi-integrable (in all variables t, x and v) or not. Proof. The method of proof of this lemma is similar to the strategy used in the proof of Lemma 7.8. An admissible renormalization ˇ1 .G˙ /. As in the proof of Lemma 7.8, we consider first the admissible square root renormalization p ˇ1 .z/ D

z C a 1 ;

for some given 1 < a < 4. Similarly to (7.16), as a consequence of the entropy and entropy dissipation bounds, we have now that p a G˙ C a 1 ; 2 gO ˙ D O 2 1 1 L .dt dxdv/ p a G˙ C a 1 ; 2 gO ˙ D O a1 L1 .dt dxdv/ C O 2 1 L dt IL2 .Mdxdv/ p a G˙ C a 1 ; 2 gO ˙ D O a1 L1 .dt dxdv/ C O 2 2 Lloc dt dxIL2 1Cjvj2 Mdv (7.32) so that strong compactness properties of p

G C a 1

p

in Lloc .dtdxdv/, for any given 1 p 2, will entail similar properties on gO ˙ in the same space and vice versa. Compactness of ˇ1 .G˙ / in v. In particular, since, in view of Lemma 7.3, the renormalized fluctuations gO ˙ are locally relatively compact in v in L2loc .dtdxdv/, the same holds true for any subsequence of p

G˙ C a 1 ;

in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup ˇ1 G˙ .t; x; v C h/ ˇ1 G˙ .t; x; v/L2 .K;dt dxdv/ < :

(7.33)

>0

Action of the transport operator on ˇ1 .G˙ /. Thus, using ˇ1 .z/ to renormalize the Vlasov–Boltzmann equations in (4.35) and decomposing the collision integrands


223

according to (5.8), we find that (see (6.9))

p G˙ G˙ C a 1 .@t C v rx ˙ ı .E C v ^ B / rv / ıE v p 2 G˙ C a p Z q G˙ ˙q D p G O˙ M dv d 2 G˙ C a R3 S2 Z ˙ 2 2 C p qO M dv d ˙ a 8 G C R3 S2 p Z q ı G˙ ˙; C p G qO M dv d 2 G˙ C a R3 S2 Z ˙; 2 2 C p M dv d: qO 8 G˙ C a R3 S2

It follows, employing the uniform bounds gO ˙ 2 L1 .dtI L2 .Mdxdv// and qO ˙ ; qO ˙; 2 L2 .MM dtdxdvdv d / from Lemmas 5.2 and 5.3, respectively, that p G˙ C a 1 .@t C v rx / p p ˙ ı G G˙ C a 1 D˙ p E v 1 C gO ırv .E C v ^ B / 2 2 G˙ C a p p Z Z G˙ G˙ ˙ ˙ qO ˙ M dv d C p gO qO M dv d C p 2 G˙ C a R3 S2 4 G˙ C a R3 S2 Z ˙ 2 2 C p qO M dv d ˙ a 8 G C R3 S2 p Z ı G˙ qO ˙; M dv d C p 2 G˙ C a R3 S2 p Z ı G˙ ˙; C p gO qO M dv d ˙ a 3 2 4 G C R S Z ˙; 2 2 C p M dv d qO ˙ a 3 2 8 G C R S D O.1/L1 .dt dxdv/ C O.ı/L1 dt dxIW 1;1 .dv/ : loc

loc

loc

(7.34) Compactness of ˇ1 .G˙ / in .x; v/. On the whole, we have established the compactness in velocity of ˇ1 .G˙ / in (7.33) and a bound on the transport operator acting on ˇ1 .G˙ / in (7.34). Therefore, a direct application of Lemma 7.7 yields that p G˙ C a 1 ˙ ˇ1 .G / D is locally relatively compact in .x; v/ in L1loc .dtdxdv/.

224


Combining this result with (7.32), implies that the renormalized fluctuations gO ˙ are relatively compact in .x; v/ in L1loc .dtdxdv/ as well. Moreover, since the families gO ˙ are uniformly bounded in L1 .dtI L2 .Mdxdv//, we easily deduce that the renormalized fluctuations gO ˙ are relatively compact in .x; v/ in Lploc .dtdxdv/, for any 1 p < 2, in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h; l 2 R3 satisfy jhj C jlj < , then sup gO ˙ .t; x C h; v C l/ gO ˙ .t; x; v/Lp .K;dt dxdv/ < ; >0

which concludes the proof of the first part of the lemma. An admissible renormalization ˇ2 .G˙ /. We proceed now to showing that, for any

> 0, the families jgO ˙ j2 1nıjgO ˙ j1o

are equi-integrable (in all variables t, x and v), which will also be seen as a consequence of Lemma 7.7. To this end, we consider now the admissible renormalization p ˇ2 .z/ D

z C a 1

2 p z C a 1 ı

;

where .z/ 2 C 1 .R/ is a cutoff satisfying 1Œ1;1 .z/ 1Œ2;2 , for some given 1 < a < 4 (in fact, we will further restrict the range of a so that necessarily a D 2 below) and any small enough > 0. Next, writing ˇ2 .z/ D ˇ1 .z/2 .ı ˇ1 .z// and using (7.23), it is readily seen that (7.32) implies that ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ4ˇ2 G ˙ gO ˙2 ı gO ˙ ˇ C ˇ2ˇ1 G ˙ gO ˙ ˇ ˇˇ1 G ˙ ˇ C ˇgO ˙ ˇ ˇ ˇ 2

a ; D O a1 C 2 1 1 1 L

dt ILloc dxIL .Mdv/

(7.35) so that the equi-integrability of ˇ2 .G˙ /

D

!2 !! p p G˙ C a 1 G˙ C a 1 ı

will clearly entail the equi-integrability of ˇ ˙ ˇ2 ˇgO ˇ 1n ˇˇ ˙ ˇˇ o ; ıˇgO ˇ1

for any > 0. Compactness of ˇ2 .G˙ / in v. Furthermore, note that the velocity compactness stated in (7.33) implies a corresponding property for ˇ2 .G˙ /. Indeed, using (7.23)


225

again, we deduce, for any h 2 R3 , that ˙ ˇ2 G .t; x; v C h/ ˇ2 G ˙ .t; x; v/ 1 Lloc .dt dxdv/ ˙ ˙ C ˇ1 G L2 .dt dxdv/ ˇ1 G .t; x; v C h/ ˇ1 G˙ .t; x; v/L2

loc .dt dxdv/

loc

:

It then follows from (7.33) that any subsequence of ˇ2 .G˙ / is locally relatively compact in v in L1 .dtdxdv/ in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup ˇ2 .G˙ /.t; x; v C h/ ˇ2 .G˙ /.t; x; v/L1 .K;dt dxdv/ < : (7.36) >0

However, in order to use the above velocity compactness of ˇ2 .G˙ / in Lemma 7.7, we still need to show that ˇ2 .G˙ / enjoys an improved integrability with respect to the velocity variable. To this end, we introduce the decomposition !! p p ˙ C a 1 ˙ C a 1 1 G G ˇ2 .G˙ / D …gO ˙ ı

2 !! !p p p G˙ C a 1 1 ˙ G˙ C a 1 G˙ C a 1 gO ı

C 2 !! p p 1 G˙ C a 1 G˙ C a 1 C .gO ˙ …gO ˙ / ı

: 2 Since …gO ˙ belongs to L1 .dtI L2 .dxI Lp .Mdv///, for any 1 p < 1, we therefore get, for any 1 r < 2, ˇp ˇ ˇ G ˙ C a 1 1 ˇ C ˇ ˇ ˇ2 .G˙ / O.1/L1 dt dxILr .Mdv/ C gO ˙ ˇ ˇ ˇ loc

ı 2 ˇ ˇ C ˇˇ ˙ C gO …gO ˙ ˇ :

ı Then, by (7.32) and the relaxation estimate (5.11), we obtain, provided 2 a < 4, ˇ2 .G˙ / D O.1/L1 dt dxILr .Mdv/ loc ! a 2 CO

ı

CO


1 DO ;

L1loc dt dxILr .Mdv/ for any 1 r < 2.

ı L1loc dt dxIL2 .Mdv/

226


Finally, combining the preceding estimate with the compactness estimate (7.36), we deduce, for any 1 r < 2, that ˇ2 .G˙ / is locally relatively compact in v in L1 .dtdxI Lr .dv//, in the sense that, for any > 0 and every compact subset K Œ0; 1/ R3 R3 , there exists > 0 such that, if h 2 R3 satisfies jhj < , then sup ˇ2 .G˙ /.t; x; v C h/ ˇ2 .G˙ /.t; x; v/ 1K .t; x; v/L1 dt dxILr .dv/ < : >0

(7.37) Action of the transport operator on ˇ2 .G˙ /. Thus, using ˇ2 .z/ to renormalize the Vlasov–Boltzmann equations in (4.35), decomposing the collision integrands according to (5.8), and writing for convenience .z/ D z 2. ız/ C ız 0 . ız/ ; so that ˇ20 .z/ D ˇ10 .z/ .ˇ1 .z//, we have now that .@t C v rx ˙ ı .E C v ^ B / p

rv /ˇ2 .G˙ /

G˙ ˇ1 .G˙ / ıE v p 2 G˙ C a

Z q G˙ ˇ1 .G˙ / ˙q D G O ˙ M dv d p 3 2 2 G˙ C a R S Z 2 ˙ ˙ 2 ˇ1 .G / C p qO M dv d 8 G˙ C a R3 S2 p Z q ı G˙ ˇ1 .G˙ / ˙; G qO M dv d C p 2 G˙ C a R3 S2 Z ˙; 2 2 ˇ1 .G˙ / C p M dv d: qO 8 G˙ C a R3 S2

It follows, employing the uniform bounds gO ˙ 2 L1 .dtI L2 .Mdxdv// and qO˙ ; qO ˙; 2 L2 .MM dtdxdvdv d / from Lemmas 5.2 and 5.3, respectively, and the direct estimates ˇ ˇ ˇ ˇ ˇ ˇ1 .G ˙ / ˇ C ˇˇ1 G ˙ ˇ D O .1/ 2 Lloc .dt dxdv/ ; 1 ˇ1 .G˙ / D O ;

ı L1 .dt dxdv/ ˇ ˇ ˇ ˇ ˇˇ2 .G ˙ /ˇ C ˇˇ1 G ˙ ˇ D O 1 ;

ı

ı L2loc .dt dxdv/


227

that, provided 1 < a 2, .@t C v rx /ˇ2 .G˙ / p

ı G˙ ˇ1 .G˙ / D˙ E v 1 C gO ırv .E C v ^ B / ˇ2 .G˙ / p 2 2 G˙ C a p Z G˙ ˇ1 .G˙ / p qO ˙ M dv d C 2 G˙ C a R3 S2 p Z G˙ ˇ1 .G˙ / ˙ ˙ p gO qO M dv d C ˙ a 3 2 4 G C R S Z ˙ 2 2 ˇ1 .G˙ / C p qO M dv d ˙ a 3 2 8 G C R S p Z ı G˙ ˇ1 .G˙ / qO ˙; M dv d C p 2 G˙ C a R3 S2 p Z ı G˙ ˇ1 .G˙ / ˙; gO qO M dv d C p 4 G˙ C a R3 S2 Z ˙; 2 2 ˇ1 .G˙ / C p M dv d qO ˙ a 8 G C R3 S2 1 DO : 1;1

L1loc dt dxIWloc .dv/ (7.38) Note here that the critical term in (7.38) preventing a better control on the transport acting on ˇ2 .G˙ / and, thereby, on the concentrations of jgO ˙ j2 , is precisely ırv .v ^ B / ˇ2 .G˙ /: Equi-integrability of ˇ2 .G˙ / in .t; x; v/. On the whole, we have established the compactness in velocity of ˇ2 .G˙ / in (7.37) and a bound on the transport operator acting on ˇ2 .G˙ / in (7.38). Therefore, further noticing that ˇ2 .G˙ / is non-negative and, recalling gO ˙2 2 L1 .dtI L1 .Mdxdv// and the error estimate (7.35), that the R family K ˇ2 .G˙ /dxdv is equi-integrable, for any compact set K R3 R3 , a direct application of Lemma 7.7 yields that !2 !! p p ˙ G˙ C a 1 G˙ C a 1 ı

ˇ2 G D is equi-integrable in all variables .t; x; v/.

228


Finally, combining this result with (7.35) one concludes that the renormalized fluctuations ˇ ˙ ˇ2 ˇgO ˇ 1n ˇˇ ˙ ˇˇ o ; ıˇgO ˇ1

for any > 0, are equi-integrable in all variables .t; x; v/ as well, which completes the proof of the lemma. Remark. Note that the estimates for one species (7.27), (7.28) and (7.29), which were deduced directly from Lemma 7.8, are no longer valid for two species in the settings of weak or strong interactions considered here. Notice, however, that the control (5.28) on the very large values of fluctuations holds in all cases, for it is a mere consequence of the entropy bound only. Thus, on the one hand, estimate (5.28) implies, for any 1 > 0 small enough, that ˇ ˙ ˇ2 1 ˇgO ˇ 1n ˇˇ ˙ ˇˇ o D O ; 1 ˇgO ˇ>1 jlog 1 j L1 dt IL1 .Mdxdv/ while, on the other hand, we showed in Lemma 7.10 by controlling the action of the transport operator on the fluctuations that, for any 2 > 0, ˇ ˙ ˇ2 ˇgO ˇ 1n ˇˇ ˙ ˇˇ o is equi-integrable (in all variables t, x and v). ı ˇgO ˇ1 2

ˇ ˇ2 In order to establish the equi-integrability of ˇgO ˙ ˇ , there would therefore remain only to control the quantity ˇ ˙ ˇ2 ˇgO ˇ 1n 1 ˇˇ ˙ ˇˇ 1 o ; 0, 1 C gO C gO L2 .Mdv/ h h C C C D 1 C gO gO 1 C gO gO 1 C gO gO 2 2 2

L .Mdv/

1 C jnj * H 1 C jnj

L .Mdv/

L .Mdv/

in w-L1loc dtdxI w-L1 ..1 C jvj/Mdv/ ;

and, similarly, 1 C gO C gO L2 .Mdv/ hO hO C C C D 1 C gO gO L2 .Mdv/ 1 C gO gO L2 .Mdv/ 1 C gO gO L2 .Mdv/ *

1 C jnj O H 1 C jnj

in w-L2loc dtdxI w-L2 .Mdv/ :


231

Therefore, for any '.v/ 2 Cc1 .R3 / such that j'.v/j .1 C jvj2 /, we find that 1 C jnj inf kh kL1 dt dxIL1 1Cjvj2 Mdv ; lim 1 C jnj H ' 1 !0 loc L dt dxIL1 .Mdv/ loc

and

1 C jnj O 1 C jnj H

inf hO lim !0


;


so that, in view of the bound on h from Lemma 5.13 and the bound on hO from Lemma 5.11 and by the arbitrariness of and ', we conclude that .1 C jnj/ H 2 L1loc dtdxI L1 1 C jvj2 Mdv ; and

.1 C jnj/ HO 2 L1loc dtdxI L2 .Mdv/ :

The justification of the lemma is complete.

Chapter 8

Higher-order and nonlinear constraint equations In Chapter 6, using weak compactness methods, we have derived lower-order linear macroscopic constraint equations for one species and for two species in a weak interaction regime. For the one-species case considered in Theorem 4.5, this is sufficient to obtain all constraint equations contained in the limiting system (4.31). As for the two-species case considered in Theorems 4.6 and 4.7, the corresponding limiting systems (4.39) and (4.38), respectively, contain higher-order constraint equations (appearing as singular perturbations of the equations of motion) and nonlinear constraint equations, namely the (solenoidal) Ohm’s law and the internal electric energy constraint, which cannot be deduced solely from the weak compactness bounds established in Chapter 5. We address now these singular limits employing the strong compactness bounds obtained in Chapter 7.

8.1 Macroscopic constraint equations for two species, weak interactions As seen in Section 6.2 (see (6.10) in the proof of Proposition 6.2), it is possible to derive limiting kinetic equations of the type Z q ˙ M dv d; (8.1) v rx g ˙ D R3 S2

from (4.35) when ı D o.1/. What we intend to do next is to take advantage of the symmetries of the collision integrands q ˙ and q ˙; and of the strong compactness bounds from Chapter 7 to go one order further and, thus, to derive a singular limit in the regime considered in Theorem 4.6. This singular limit is precisely the content of Proposition 8.1, which will eventually yield the solenoidal Ohm’s law and the internal electric energy constraint from (4.39) in Proposition 8.2 below. Of course, since we are considering renormalized fluctuations, we do not expect that the integrals in v of the right-hand sides of the Vlasov–Boltzmann equations in (4.35) against collision invariants are zero, but they should converge to zero as ! 0 provided that we choose some appropriate renormalization which is sufficiently close to the identity. To estimate the ensuing conservation defects, we will also need to truncate large velocities. Note that, even if conservation laws were known to hold for renormalized solutions of (4.35), we would have to introduce similar truncations of large tails and large velocities in order to control uniformly the flux and acceleration terms.

234

8 Higher-order and nonlinear constraint equations

The main result in this section concerning the derivation of higher-order nonlinear constraint equations in the regime considered in Theorem 4.6 is contained in the following proposition. Proposition 8.1. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.6 for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded. In accordance with Lemmas 5.1, 5.2 and 5.3, denote by q ˙; 2 L2 .MM dtdxdvdv d /; g ˙ 2 L1 dtI L2 .Mdxdv/ ; and E; B 2 L1 dtI L2 .dx/ ; any joint limit points of the families gO ˙ and qO ˙; defined by (5.3) and (5.6), E and B , respectively. Then, one has Z ˙ q ˙; vMM dvdv d D rx pN .E C u ^ B/ ; R3 R3 S2 2 (8.2) Z 5 ˙; jvj q MM dvdv d D 0; 2 2 R3 R3 S2 where u is the bulk velocity associated with the limiting fluctuations g ˙ and pN 2 L1loc .dtdx/ is a pressure. The proof of Proposition 8.1 is lengthy and contains several steps. Therefore, for the sake of clarity, it is deferred to Section 8.1.1 below. As a direct consequence of the previous proposition, we derive in the next result the solenoidal Ohm’s law and the internal electric energy constraint from (4.39). Proposition 8.2. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.6 for weak interspecies interactions, i.e., ı D o.1/ and ı unbounded. In accordance with Lemmas 5.1, 5.2, 5.13 and 5.14 denote by h 2 L1loc dtdxI L1 .1 C jvj2 /Mdv ; g ˙ 2 L1 dtI L2 .Mdxdv/ ; and E; B 2 L1 dtI L2 .dx/ ; any joint limit points of the families gO ˙ and h defined by (5.3) and (5.29), E and B , respectively. Then, one has j D .rx pN C E C u ^ B/

and

w D 0;

where u is the bulk velocity associated with the limiting fluctuations g ˙ , j and w are, respectively, the electric current and the internal electric energy associated with the limiting fluctuation h, pN 2 L1loc .dtdx/ is a pressure and the electric conductivity > 0 is defined by (2.75).


235

Proof. By Proposition 8.1, we have that Z ˙ q ˙; vMM dvdv d D rx pN .E C u ^ B/ ; 3 3 2 R R S 2 Z 5 ˙; jvj q MM dvdv d D 0: 2 2 R3 R3 S2 Then, further incorporating identity (6.11) from Proposition 6.3 into the above relations yields that 2 Z Z 1 1 jvj vM dv D rx p; E Cu^B j L .v/ vM dv wL N 2 R3 2 R3 2 2 2 2 Z Z 5 jvj 5 jvj jvj M dv C M dv D 0: j L .v/ wL 2 2 2 2 2 R3 R3 2 2 R R Since,Rby symmetry considerations, R3 L.v/ jvj2 52 M dv D R3 L jvj2 vM dv D 0 and R3 L.vi /vj M dv D 0, if i ¤ j , we compute that E Cu^B

1 j D rx pN

and

1 w D 0;

where > 0 and > 0 are defined in (2.75) and (2.76), respectively, which concludes the proof of the proposition.

8.1.1 Proof of Proposition 8.1 Here, we analyze the equations (4.37), which have to be renormalized, at a higher order. This becomes more complicated than the previous asymptotic analysis of (6.9) in the proof of Proposition 6.2, because we do not have enough strong compactness to take limits in the nonlinear terms p G˙ C a 1 ˙ .v ^ B / rv therein. More precisely, we are not able to control the concentrations of jgO ˙ j2 (see Lemma 7.10). We will therefore consider a stronger renormalization of the equation for the fluctuations of density and exploit the equi-integrability from Lemma 7.10, however weak it may be. 8.1.1.1 An admissible renormalization We introduce the admissible renormalization .z/ defined by z 1 ; .z/ 1 D .z 1/ ı

236


where

ı

1 is small and 2 C 1 .R/ satisfies 1Œ1;1 .z/ .z/ 1Œ2;2 .z/;

for all z 2 R:

Without distinguishing, for simplicity, the notation for cations and anions, we denote D ıg˙ ; O D 0 G˙ : Thus, renormalizing the Vlasov–Boltzmann equation from (4.35) with respect to .z/ yields

1 @t C v rx ˙ .E C v ^ B / rv g˙ E vG˙ O ı ı 1 ı D 2 O Q G˙ ; G˙ C 2 O Q G˙ ; G : ı

Notice here that there are two singular terms in the equations above, namely 1ı v rx g˙ and the first term in the right-hand side (as shown below, the second term in the right-hand side is not singular). Therefore, in order to annihilate asymptotically these singular expressions, we integrate now the above equations against

jvj2 '.v/ Kı

Mdv;

with Kı D K jlog ıj, for some large K > 0 to be fixed later on, for any collision invariant '.v/ and some smooth compactly supported truncation 2 Cc1 .Œ0; 1// satisfying 1Œ0;1 1Œ0;2 . This leads to 2 2 Z 1 jvj jvj M dv C rx vM dv g˙ ' g˙ ' 3 3 K ı Kı ı R R 2 Z jvj ˙ M dv g .E C v ^ B / rv ' Kı R3 2 Z jvj vM dv 1 C g˙ O ' E 3 Kı R 2 Z ˙ ˙ 1 jvj M dv O Q G ; G ' D 2 ı R3 Kı 2 Z ı jvj M dv: O Q G˙ ; G ' C 2 R3 Kı

@t ı

Z

(8.3)


237

8.1.1.2 Convergence of conservation defects Let us focus on the right-hand side of (8.3) first. One has 2 Z ˙ ı jvj M dv O Q G ; G ' 2 R3 Kı 2 Z q jvj ˙; (8.4) MM dvdv d O G˙ G qO ' D Kı R3 R3 S2 2 Z ˙; 2 2 jvj MM dvdv d: O qO ' C 4ı R3 R3 S2 Kı Since

ı

vanishes,

jvj2 Kı is bounded pointwise by a constant multiple of jlog ıj and the collision integrands qO ˙; are uniformly bounded in L2 .MM dtdxdvdv d /, according to Lemma 5.3, we find that the second term in the right-hand side of (8.4) vanishes in L1 .dtdx/. Further utilizing that, thanks to Lemma 5.2, q G˙ D 1 C O./L2 dt IL2 .Mdxdv/ ; loc q ; ˙ D 1 C O./ G L2 dt IL2 M dxdv

'

loc

p

and that O G˙ is uniformly bounded pointwise, it is readily seen that the weak limit of the first term in the right-hand side of (8.4) coincides with the weak limit of 2 Z jvj MM dvdv d; O qO˙; ' 3 3 2 Kı R R S which, since

jvj2 K is dominated by j'j and converges almost everywhere to ', is easily shown to converge weakly in L2 .dtdx/ to Z q ˙; 'MM dvdv d:

O '

R3 R3 S2

Thus, so far, we have established that the second term in the right-hand side of (8.3) satisfies 2 Z Z ˙ ı jvj M dv * O Q ; G q ˙; 'MM dvdv d G ' 3 3 2 2 R3 Kı R R S in L1loc .dtdx/: (8.5)

238


The first term in the right-hand side of (8.3) is more singular and, therefore, harder to control. One has, in this case, taking advantage of collisional symmetries, that 2 Z ˙ ˙ 1 jvj M dv O Q ; G G ' ı 2 R3 Kı 2 Z 2 jvj ˙ 2 qO MM dvdv d O ' D 4ı R3 R3 S2 Kı 2 q Z 1 jvj ˙ MM dvdv d qO ˙ G˙ G O ' 1 ı R3 R3 S2 Kı Z q

1 (8.6) ˙ MM dvdv d ' qO˙ G˙ G O 1 O C ı R3 R3 S2 Z q

1 0 ˙ MM dvdv d O O C 1 O0 O ' qO ˙ G˙ G ı R3 R3 S2 Z 2 2 0 0 O O O O ' qO ˙ MM dvdv d 4ı R3 R3 S2 def

D D1 .'/ C D2 .'/ C D3 .'/ C D4 .'/ C D5 .'/;

where we have used that ' is a collision invariant to symmetrize the last term. Now, we control each term Di .'/, i D 1; : : : ; 5, separately. The vanishing of the first term D1 .'/ for any function '.v/ growing at most quadratically at infinity easily follows, using Lemma 5.3, from the estimate 2 2 1 2 jvj ˙ D .'/ 1 qO 2 ' L MM dt dxdvdv d O 1 1 L .dt dx/ L 4ı Kı L 2 2 C Kı D CK j log ıj: ı ı (8.7) The second term D2 .'/ is controlled by the following estimate on the tails of Gaussian distributions: for any p 2 R, as R ! 1, r Z 2 pC1 R p (8.8) jvj M.v/ dv R 2 e 2; fjvj2 >Rg in the sense that the quotient of both sides converges to 1 as R ! 1, which is easily established by applying the Bernoulli–l’Hospital rule. We have indeed ˇ 2 ˇ ˇD .'/ˇ q 1 ˙ ˙ ˙ qO L2 MM dvdv d O '1fjvj2 Kı g G G 2 ı L MM dvdv d q q C ˙ ˙ G˙ ' qO L2 MM dvdv d O G 1 2 1 fjvj Kı g L2 .Mdv/: ı 2 L L .Mdv/


239

p Thus, using the bound from Lemma 5.3, the pointwise boundedness of 0 .z/ z, and the Gaussian decay estimate (8.8), we get, for all '.v/ growing at most quadratically at infinity, K

5 D2 .'/ D O ı 4 1 jlog ıj 4 1 ; (8.9) Lloc .dt dx/

which tends to zero as soon as K > 4. The last term D5 .'/ is mastered using the same tools. For high energies, i.e., when jvj2 Kj log ıj, we obtain D5> .'/ 2Z 2 def 0 0 D O O O O '1fjvj2 Kı g qO ˙ MM dvdv d 4ı R3 R3 S2 2 q 2 ˙ 1 ˙ ; O G O 1 qO L2 MM dvdv d '1fjvj2 Kı g 2 L L MM dvdv d ı L1 so that, using the estimate (8.8) on the tails of Gaussian distributions and the bound on qO ˙ from Lemma 5.3, K

5 D5> .'/ D O ı 4 1 jlog ıj 4 2 ; (8.10) L .dt dx/

which tends to zero as soon as K > 4. For moderate energies, i.e., when jvj2 < Kj log ıj, we easily find 2 Z 2 def 5< 0 0 O O O O '1fjvj2 1o ı ı ı ˇˇ ˇ ˇˇ ˇ ˇ ˇˇ ˇ ˇ ˇ ˙ ˇ nˇ ˇ ˇ ˙ ˇ nˇ ˇ o ˇ1 O ˇ g 1 ˇˇgO ˙ ˇˇ1 C ˇ1 O ˇ gO 1 ˇˇgO ˙ ˇˇˇ>1o ı ˇ ˇ ˇ ˇ ˇ ˙ ˙2 ˇ nˇ C ˇgO C gO ˇ 1 ˇˇgO ˙ ˇˇˇ1o C C ˇgO ˙ ˇ 4 ı ˇˇ ˙ ˇˇ C C gO : ı

(8.12)

Therefore, thanks to the bound on qO from Lemma 5.3, we infer that 3 D .'/ 1 L

loc .dt dx/

Thus, we conclude that

: gO ˙ 2 C C L 2 ı loc dt dxIL .Mdv/

3 D .'/ 1 L

loc .dt dx/

C :

(8.13)

A similar argument provides the convergence of the remaining term D4 .'/. Thus, one has by the Cauchy–Schwarz inequality, for any 2 < p < 1, ˇ 4 ˇ ˇD .'/ˇ

q 1 ˙ 0 0 ˙ ˙ qO L2 MM dvdv d O O 1 O O ' G G 2 ı L MM dvdv d q 2

1 ˙ 0 0 ˙ 1 O O ' C qO L2 MM dvdv d O G 2 L1 ı L MM dvdv d

1 ˙ Cp qO L2 MM dvdv d 1 O : ı p L .Mdv/

Therefore, thanks to the bound on qO ˙ from Lemma 5.3, we infer, for any 2 < p < 1, 4 D .'/ 1 L

loc .dt dx/

1 C 1 O : ı L2 dt dxILp .Mdv/ loc


241

Next, using estimate (5.11) from Lemma 5.10 and the bound (8.12), we find that ˇ2 1 ˇˇ ˇ O ˇ1 ˇ ı2 ˇ ˇˇ ˙ ˇˇ 1 ˇˇ ˇ C C gO ˇ1 O ˇ ı ı ˇ ˇ 1 ˇˇ ˇˇ ˙ ˇˇ ˇˇ ˙ ˇ …gO C gO …gO ˙ ˇ ˇ1 O ˇ C C ı ı ˇ 2 ˇˇ ˙ ˇˇ ˇˇ ˙ ˇˇ 1 ˇˇ ˙ gO …gO ˙ ˇ C C …gO gO C C C ı ı ı 2 ; C O

C O C ı ı ı L1loc dt dxIL2 .Mdv/ r L1 loc dt dxIL .Mdv/ for any 1 r < 2. Then, we end up with 4 D .'/

L1 loc .dt dx/

C :

(8.14)

Finally, incorporating (8.7), (8.9), (8.10), (8.11), (8.13) and (8.14) into (8.6), we conclude that the first term in the right-hand side of (8.3) is uniformly bounded in L1loc .dtdx/ and satisfies, for any > 0, Z 1 lim sup 2 ı !0

R3

O Q

2 ˙ ˙ jvj M dv G ; G ' 1 Kı L

C :

(8.15)

loc .dt dx/

In particular, combining (8.5) and (8.15), it follows from the Banach–Alaoglu theorem, up to further extraction of subsequences, that the right-hand side of (8.3) converges in the weak- topology of Radon measures Mloc .Œ0; 1/ R3 / to Z R3 R3 S2

q ˙; 'MM dvdv d C Q .'/;

(8.16)

where the Radon measure Q .'/ 2 Mloc .Œ0; 1/ R3 / obeys the control Q .'/

Mloc Œ0;1/R3

C :

(8.17)

242


8.1.1.3 Decomposition of flux terms Next, we treat the convergence in (8.3) of the flux terms 2 Z 1 jvj ˙ vM dv: g ' rx ı Kı R3 To this end, we use the decomposition (5.5) to write ˙ 2 ˙ 2 g˙ D gO ˙ C D …gO ˙ C gO ˙ …gO ˙ C ; gO gO 4 4 where … is the orthogonal projection on Ker L in L2 .Mdv/, which yields the decomposition of flux terms 2 Z 1 jvj ˙ rx vM dv g ' 3 ı Kı R 2 Z ˙ 2 jvj rx vM dv D gO ' 4ı Kı R3 2 Z ˙ 1 jvj ˙ vM dv C rx gO …gO ' 3 ı Kı (8.18) R 2 Z 1 jvj 1 vM dv …gO ˙ ' C rx ı Kı R3 Z 1 …gO ˙ 'vM dv C rx ı R3 def

D F1 .'/ C F2 .'/ C F3 .'/ C F4 .'/:

Then, from the condition on the support of , and since ˇ ˇ ˇ ˙ˇ ˇ ˙ ˇ ˇ g˙ ˇ ˇ ˇ ˇ ˇgO ˇ D ˇ ˇ 1 C gO ˙ ˇ 2 g ; 4 we deduce that, on the support of , ˇ ˙ˇ ˇ ˇ ˇgO ˇ 2 ˇg ˙ ˇ 4 4 :

ı Therefore, by Lemma 5.12, for any 1 p < 2, we have that .gO ˙ /2 is uniformly bounded in L1loc .dtdxI Lp .Mdv//. Hence, we conclude that kF1 .'/kW 1;1 .dt dx/ C k.gO ˙ /2 kL1 dt dxILp .Mdv/ kv'kLp0 .Mdv/ C : loc ı ı loc (8.19) Moreover, by (5.11), we easily get C ˙ kgO …gO ˙ kL1 dt dxIL2 .Mdv/ k k1 kv'kL2 .Mdv/ loc ı loc C ; ı (8.20) which handles the second term. kF2 .'/kW 1;1 .dt dx/

243


Further note that, by the definition of the projection …, we have, for any 2 < p < 1, k…gO ˙ kL1 dt IL2 dxILp .Mdv/ Cp kgO ˙ kL1 dt IL2 .Mdxdv/ ; whence kF3 .'/kW 1;1 .dt dx/

1 kv'kLq .Mdv/ ı 2 2 L .dt dxIL .Mdv// loc 2 1 jvj C C kgO ˙ kL1 dt IL2 .Mdxdv/ ; v' 1 2 ı Kı L .Mdv/ loc

Cp kgO ˙ kL1 dt IL2 .Mdxdv/

with q1 D 12 p1 . Then, using estimate (8.12) (with instead of O ) and the control of Gaussian tails (8.8) to respectively bound the first and second terms in the righthand side above, we infer that K

7

kF3 .'/kW 1;1 .dt dx/ C C C ı 4 1 j log ıj 4 ; loc

which is small provided that K > 4. Therefore, up to further extraction of subsequences, we deduce that F3 .'/ * rx R .'/ in the sense of distributions, where the Radon measure R .'/ 2 Mloc .Œ0; 1/ R3 / satisfies the control C : R .'/ 3 Mloc Œ0;1/R

(8.21)

(8.22)

The form of the last remaining flux term F4 .'/ depends on the collision invariant ': 2

If '.v/ D v, we get, using that .v/ D v ˝ v jvj3 Id is orthogonal to the collision invariants, Z 1 F4 .'/ D rx …gO ˙ v ˝ vM dv ı R3 Z 1 (8.23) …gO ˙ jvj2 M dv D rx 3ı R3

1 D rx O˙ C O˙ ; ı where O˙ and O˙ denote the densities and temperatures, respectively, associated with the renormalized fluctuations gO ˙ . Thus, this term takes the form of a gradient and will therefore vanish upon integrating it against divergence free vector fields, as required by the theory of weak solutions of Leray.

244


If '.v/ D

jvj2 2

52 , we obtain

F4 .'/ for

.v/ D

jvj2 2

1 D rx ı 5 2

Z R3

…gO ˙

jvj2 5 vM dv D 0; 2 2

(8.24)

v is orthogonal to the collision invariants.

Thus, on the whole incorporating (8.19), (8.20), (8.21), (8.23) and (8.24) into (8.18), we conclude that the flux terms satisfy the following convergences in the sense of distributions: 2 Z

1 jvj ˙ P v ˝ vM dv * P rx R .v/ ; g rx ı Kı R3 2 2 2 (8.25) Z 1 5 5 jvj jvj ˙ jvj g rx vM dv * rx R ; ı Kı 2 2 2 2 R3 where P denotes the Leray projector onto solenoidal vector fields. 8.1.1.4 Decomposition of acceleration terms It only remains to deal with the terms involving the electromagnetic field in (8.3), which we decompose as 2 Z jvj ˙ M dv g .E C v ^ B / rv ' 3 Kı R Z jvj2 vM dv 1 C g˙ O ' C E Kı R3 2 2 Z Z jvj jvj ˙ ˙ M dv C E vM dv g rv ' g O ' D E Kı Kı R3 R3 Z Z 2 jvj O 1 'vM dv 'vM dv C E C E Kı R3 R3 Z g˙ v ^ rv 'M dv B 3 R 2 Z jvj ˙ 1 M dv g v ^ .rv '/ C B Kı R3 def

D A1 .'/ C A2 .'/ A3 .'/: (8.26) From the condition on the support of and since ˇ ˇ ˇ ˙ ˇ ˇ g˙ ˇ ˇ ˇ ˇgO ˇ D ˇ ˇ 2 ˇg ˙ ˇ ; ˇ 1 C gO ˙ ˇ 4


245

we clearly have that kA1 .'/kL1 dt IL1 .dx/

2 ˙ ˇ o nˇ ' jvj v CkE kL1 .dt IL2 .dx// g 1 ˇˇıg ˙ ˇˇ2 1 Kı L1 .dt IL2 .Mdxdv// H .Mdv/ ˙ ˙2 ˇ o nˇ C gO C 4 gO 1 ˇˇı gO ˙ ˇˇ4 1 L .dt IL2 .Mdxdv// ˙ C gO ; 1 2

L

.dt IL .Mdxdv//

(8.27) which handles the first acceleration term. Then, in order to deal with the second acceleration term, we estimate first, using (8.12) and the control of Gaussian tails (8.8), that Z 2 jvj O 1 'vM dv 3 K 2 ı R Lloc .dt dx/ Z Z 2

jvj 1 'vM dv C O 1 'vM dv 2 Kı R3 R3 L2 Lloc .dt dx/ loc .dt dx/ ˙ K 2 C ı gO L2 .dt dxIL2 .Mdv// C C ı 2 j log ıj ; loc

whence, as ! 0, A2 .'/

Z

* E

'vM dv

in L1 dtI L2 .dx/ :

(8.28)

R3

As for the remaining term A3 .'/, note first, using (8.8) again, that 2 Z jvj ˙ B 1 M dv g v ^ .r '/ v 1 Kı R3 L .dt IL2 .dx// Z 2 C C K 3 jvj 1 jvj2 M dv ı 2 1 j log ıj 2 ; kB kL1 .dt IL2 .dx// 3

ı K

ı R which is small as soon as K > 2. Moreover, in view of (5.5), we find that Z ˙ ˙ B g gO v ^ rv 'M dv R3

Z D B

R3

C

˙2 gO v ^ rv 'M dv 4

kB kL2 loc

ı

L1 loc .dt dx/

L1 loc .dt dx/

˙ ; O 2 L .dt dx/ g dt dxIL2 .Mdv/ loc

246


so that, overall, the weak limit of A3 .'/ will coincide with the weak limit of Z gO ˙ v ^ rv 'M dv: B R3

In order to take the weak limit of the preceding term, notice, in view of Lemma 7.10 a straighforward application through of the meanvalue theorem to the function z ı z C 4 z 2 , that gO ˙ D gO ˙ ı gO ˙ C 4 gO ˙2 is locally relatively compact in .x; v/ in Lp .dtdxdv/, for any 1 p < 2. In fact, Lemma 7.10 further implies that ˇ ˇ ˇ ˙ ˇ2 ˇˇ ˙ ˇˇ2 n ˇ ˇ o ˇgO ˇ gO 1 ı ˇˇgO ˙ ˇˇ4

gO ˙

is equi-integrable. Therefore, we conclude that is locally relatively compact in approximate gO ˙ , .x; v/ in L2 .dtdxdv/. In particular, for any fixed > 0, one can uniformly in > 0, in L2loc .dtdxdv/ by its regularized version gO ˙ x;v a , where a > 0 and a .x; v/ D a16 xa ; av is an approximate identity, with 2 Cc1 .R3 R3 / R such that R3 R3 .x; v/dxdv D 1. We use now compensated compactness in the following form. From the Faraday equation in (4.35), we deduce that @t B 2 L1 dtI H 1 .dx/ ; so that B enjoys some strong compactness with respect to the time variable. We then deduce, up to extraction of subsequences, that

B gO ˙ x;v a * Bg ˙ x;v a ; where g ˙ is the weak limit of gO ˙ , which coincides with the weak limits of gO ˙ and g˙ (note that ! 1 almost Incidentally, by the uniformity of the ˙ everywhere). ˙ approximation of gO by gO x;v a in L2loc .dtdxdv/, we infer that B gO ˙ * Bg ˙ ; in L1loc .dtdxdv/, whence A3 .'/ * B

Z R3

g ˙ v ^ rv 'M dv

in L1loc .dtdx/:

(8.29)

Thus, incorporating (8.27), (8.28) and (8.29) into (8.26), we finally conclude that 2 Z jvj ˙ M dv g .E C v ^ B / rv ' Kı R3 2 Z jvj (8.30) vM dv 1 C g˙ O ' C E Kı R3 Z Z * E 'vMdv B g ˙ v ^ rv 'M dv; R3

in L1loc .dtdx/.

R3


247

8.1.1.5 Convergence We are now in a position to pass to the limit in (8.3). To this end, note first that, since ı ! 0 and g˙ is uniformly bounded in the space L1loc .dtdxI L1 ..1 C jvj2 /Mdv// by Lemma 5.1, the density term @t ı

Z R3

g˙ '

jvj2 Kı

M dv;

vanishes as ! 0 and thus brings no contribution to the weak limit. Therefore, according to the weak limits of conservation defects (8.16), flux terms (8.25) and acceleration terms (8.30), we conclude, letting ! 0 in (8.3) in the sense of distri2 butions for the collision invariants ' D v and ' D jvj2 52 , that P rx R .v/ .E C u ^ B/ Z q ˙; vMM dvdv d C Q .v/ ; DP R3 R3 S2 2 5 jvj rx R 2 2 2 2 Z 5 5 jvj jvj q ˙; MM dvdv d C Q ; D 2 2 2 2 R3 R3 S2 where u denotes the bulk velocity associated with the limiting fluctuations g ˙ (recall that, according to Lemmas 5.10 and 5.11, g C and g are infinitesimal Maxwellians which differ only by their densities). Next, in view of the bounds (8.17) and (8.22) on the Radon measures Q and R , respectively, we deduce, by the arbitrariness of > 0, that Z ˙; P E Cu^B ˙ q vMM dvdv d D 0; R3 R3 S2 2 Z 5 jvj q ˙; MM dvdv d D 0; 2 2 R3 R3 S2 which concludes the proof of Proposition 8.1.

8.2 Macroscopic constraint equations for two species, strong interactions We move on now to the regime of strong interspecies interactions considered in Theorem 4.7. In this setting, the derivation of even the simplest macroscopic constraint equations (such as the incompressibility and Boussinesq contraints) involves the handling of nonlinear terms. Indeed, the limiting kinetic equation (8.1) for weak

248


interspecies interactions (obtained in Chapter 6 with weak compactness methods) corresponds now, for strong interspecies interactions, to the nonlinear equation Z ˙ .v rx ˙ .v ^ B/ rv /g ˙ E v D q C q ˙; M dv d; R3 S2

which is obtained in the proof of Proposition 8.3, below (see (8.37)), and requires the compactness properties established in Chapter 7. The next result fully characterizes the limiting kinetic equations in the regime of strong interactions. Proposition 8.3. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.7 for strong interspecies interactions, i.e., ı D 1. In accordance with Lemmas 5.1, 5.2 and 5.3, denote by g ˙ 2 L1 dtI L2 .Mdxdv/ and q ˙ ; q ˙; 2 L2 MM dtdxdvdv d any joint limit points of the families gO ˙ , qO ˙ and qO ˙; defined by (5.3) and (5.6), respectively, and by E; B 2 L1 dtI L2 .dx/ any joint limit points of the families E and B , respectively. Then, one has Z C 1 q C q C q C; C q ;C M dv d D W rx u C 2 R3 S2 and

1 2

rx ;

(8.31)

Z

C q q C q C; q ;C M dv d R3 S2 1 C D rx . / .E C u ^ B/ v; 2

(8.32)

where ˙ , u and are, respectively, the densities, bulk velocity and temperature associated with the limiting fluctuations g ˙ , and and are the kinetic fluxes defined by (2.14). Furthermore, ˙ , u and satisfy the constraints C C div u D 0; rx C D 0: (8.33) 2 In particular, the strong Boussinesq relation

C C 2

C D 0 holds.

Proof. The case ı D 1 is more complicated because we do not have enough strong compactness to take limits in the nonlinear terms p G˙ C a 1 : ˙ .v ^ B / rv


249

More precisely, we are not able to control the concentrations of jgO ˙ j2 (see Lemma 7.10). The idea is therefore to consider a stronger renormalization of the equation for the fluctuations of density. To this end, we introduce the admissible renormalization .z/ defined by z 1 .z/ 1 D .z 1/

; where > 0 is small and 2 C 1 .R/ satisfies that 1Œ1;1 .z/ .z/ 1Œ2;2 .z/;

for all z 2 R:

Without distinguishing, for simplicity, the notation for cations and anions, we denote D g˙ ; O D 0 G˙ : Thus, renormalizing the Vlasov–Boltzmann equation from (4.35) with respect to .z/ yields .@t C v rx ˙ .E C v ^ B / rv /g˙ E vG˙ O Z Z q q ˙ 2 2 ˙q D O G˙ G O˙ M dv d C O qO M dv d 4 R3 S2 R3 S2 Z Z q q ˙; 2 2 ˙; ˙ G qO M dv d C O M dv d: qO C O G 3 2 3 2 4 R S R S (8.34) Next,pemploying a strategy similar to the proof of Proposition 6.1, in particular, since O G˙ is uniformly bounded pointwise, utilizing that, thanks to Lemma 5.2, q G˙ D 1 C O./L2 dt IL2 .Mdxdv/ ; loc q ; ˙ D 1 C O./ G L2 dt IL2 M dxdv loc

and that, thanks to Lemma 5.3, the collision integrands qO˙ and qO ˙; are uniformly bounded in L2 .MM dtdxdvdv d /, we see that the weak limit of the right-hand side of (8.34) coincides with the weak limit of Z ˙ qO C qO ˙; M dv d C Q ; R3 S2

where we denote the remainder h iZ Q D O 1

R3 S2

˙ qO C qO ˙; M dv d:

250


ˇ ˇ ˇ ˇ 1r ˇ , for any given 1 r 1 and for every Then, since ˇ0 .z/ 1ˇ Cr ˇ z1 z 0, it holds that, employing the uniform bounds from Lemmas 5.1 and 5.3, for any 2 r 1, Z ˇ ˇ ˇ1 ˙ 1 ˇ ˇ ˇ qO C qO ˙; M dv d ˇQ ˇ C r ˇg˙ ˇ r R3 S2

Z

ˇ ˇ1 C ˇg˙ ˇ r 1 D O r 2r 1 r

R3 S2

˙ 2 qO C qO ˙; M dv d

2Cr Lloc .dt dxdv/

12

:

Moreover, when r D 2, it is readily seen, in view of the weak relative compactness of g˙ in L1loc .dtdxdv/ established in Lemma 5.1 and employing the Dunford– Pettis compactness criterion (see [68]), that Q is weakly relatively compact in L1loc .dtdxdv/, as well. Therefore, up to extraction of a further subsequence as ! 0, we may assume that Q converges weakly in 2r

2Cr .dtdxdv/ Lloc

to some

2r

2Cr Q 2 Lloc .dtdxdv/; 1

for any 2 r 1, whose magnitude is at most of order r . On the whole, we have evaluated that the right-hand side of (8.34) converges weakly towards Z ˙ q C q ˙; M dv d C Q R3 S2 Z 1 ˙ D : q C q ˙; M dv d C O r 2r R3 S2

2Cr Lloc .dt dxdv/

As for the left-hand side of (8.34), we first have that

E vG˙ O D E v C E v O 1 C E vg˙ O :

(8.35) 1

jr , Therefore, since g˙ O is uniformly bounded pointwise and j0 .z/1j Cr j z1 for any given 1 r 1 and for every z 0, we find, for any 2 r 1, 1 CO ; E vG˙ O D E v C O r 2r 2Cr

L2loc .dt dxdv/ Lloc .dt dxdv/


251

so that the expression from (8.35) converges weakly, for any 2 r 1, towards 1 E v C O r : 2r 2Cr Lloc .dt dxdv/

relatively compact in L1loc .dtdxdv/, by Lemma 5.1, it Next, since g˙ is weakly ˙ holds that g 1 is uniformly small in L1loc .dtdxdv/, when > 0 is small. In particular, we deduce that the family g˙ , which is weakly relatively compact in Lploc .dtdxdv/, for any 1 p < 1, converges weakly towards some g ˙; 2 Lploc .dtdxdv/ such that g ˙; D g ˙ C o.1/L1

loc .dt dxdv/

;

as ! 0. In fact, we claim that this can be improved to g ˙; D g ˙ C o.1/L2

loc .dt dxdv/

;

(8.36)

as ! 0. Indeed, up of subsequences, denoting by r ˙ 2 L2loc .dtdxdv/ ˇ ˙toˇ extraction 1 the weak limit of ˇˇg ˇˇ in Lloc .dtdxdv/, which coincides, in view of (5.5), with the weak limit of ˇgO ˙ ˇ in L2loc .dtdxdv/, it clearly holds that, for any non-negative ' 2 Cc1 .Œ0; 1/ R3 R3 /, Z ˇ ˇ ˇ ˙ ˇ ˇg g ˙; ˇ ' dtdxdv Œ0;1/R3 R3 Z ˇ

ˇ ˇ ˇ ˙ lim inf ˇg 1 ˇ ' dtdxdv !0 Œ0;1/R3 R3 Z ˇ ˙ˇ ˇg ˇ ' dtdxdv lim inf !0 Œ0;1/R3 R3 Z r ˙ ' dtdxdv; D Œ0;1/R3 R3

ˇ ˇ2 2 whence ˇg ˙ g ˙; ˇ is dominated by the integrable function r ˙ . It then follows from a direct application of Lebesgue’s dominated convergence theorem that (8.36) holds. At last, we deal with the convergence of the problematic nonlinear term B g˙ from (8.34). To this end, note first, according to (5.5) and Lemma 7.10, for any given

> 0, that the family g˙ is locally relatively compact in .x; v/ in L2 .dtdxdv/. ˙ In particular, for any fixed > 0, one can approximate ˙ g , uniformly in > 2 0, in Lloc .dtdxdv/ by its regularized version g x;v a , where a > 0 and a .x; v/ D a16 xa ; va is an approximate identity, with 2 Cc1 .R3 R3 / such that R R3 R3 .x; v/ dxdv D 1.

252


We use now compensated compactness in the following form. From the Faraday equation in (4.35), we deduce that @t B 2 L1 dtI H 1 .dx/ ; so that B enjoys some strong compactness with respect to the time variable. We then deduce, up to extraction of subsequences, that

B g˙ x;v a * Bg ˙; x;v a ; and, incidentally, by the uniformity of the approximation of g˙ by g˙ x;v a in L2loc .dtdxdv/, that B g˙ * Bg ˙; ; in L1loc .dtdxdv/. We may now take weak limits in (8.34) to infer that, for any given 2 r 1, .v rx ˙ .v ^ B/ rv /g ˙; E v Z 1 ˙ D q C q ˙; M dv d C O r

2r

2Cr Lloc .dt dxdv/

R3 S2

Finally, in view of (8.36), letting ! 0, we arrive at Z ˙ .v rx ˙ .v ^ B/ rv /g ˙ E v D q C q ˙; M dv d;

:

(8.37)

R3 S2

which, together with the fact, according to Lemmas 5.10 and 5.11, that g C and g are infinitesimal Maxwellians, which differ only by their densities C and , provides that Z ˙ q C q ˙; M dv d R3 S2 jvj2 ˙ D div . C /v C u C u C .E C u ^ B/ v 3 1 D . W rx u C rx / C rx .˙ C / .E C u ^ B/ v C .div u/ jvj2 : 3 Equivalently, we find that Z C 1 q C q C q C; C q ;C M dv d 2 R3 S2 C 1 C D . W rx u C rx / C rx C v C .div u/ jvj2 ; 2 3


and

1 2

Z

253

q C q C q C; q ;C M dv d R3 S2 1 D rx .C / .E C u ^ B/ v: 2

Then, remarking that q ˙ and q ˙; inherit the collisional symmetries of q˙ , q˙; , qO ˙ and qO ˙; , we get 0 1 Z 1 C C; ;C @ v A Cq MM dvdv d D 0; q Cq Cq R3 R3 S2

jvj2 2

so that, since .v/ and .v/ are orthogonal to the collisional invariants, the constraints (8.33) hold. The proof of the proposition is now complete. The next proposition further characterizes the limiting collision integrands. Proposition 8.4. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.7 for strong interspecies interactions, i.e., ı D 1. In accordance with Lemmas 5.1, 5.2, 5.3, 5.13, 5.14 and 7.11 denote by g ˙ 2 L1 dtI L2 .Mdxdv/ ; q ˙; 2 L2 MM dtdxdvdv d ; h 2 L1loc dtdxI L1 .1 C jvj2 /Mdv ; any joint limit points of the families gO ˙ , qO ˙; and h defined by (5.3), (5.6) and (5.29), respectively. Then, one has Z C (8.38) q q M dv d D L .h/ ; R3 S2

2 C; jvj L .h/ ; q ;C M dv d D nu L.v/ C n L q 2 R3 S2

Z

and

q C C q q C; q ;C D 0;

(8.39)

(8.40)

where n D C , u and are, respectively, the charge density, bulk velocity and temperature associated with the limiting fluctuations g ˙ .

254


Proof. We start from the decomposition Z

ˇ C ˇ2 1 ˇˇ C ˇˇ2 2 2 O ˇ ˇ h D h C gO jgO j M dv gO jgO j 4 R3

1 C D hO C gO gO nO gO C C gO C nO gO C C gO Z4 C C C 1 gO gO nO gO C gO M dv C nO O C O ; 4 R3

(8.41)

which follows from the decomposition (5.5) of fluctuations. In order to apply the compactness results from Lemmas 5.14 and 7.11, we consider the following renormalization of the above decomposition: " # nO C hO C hO 1 h D C gO C gO C gO C gO R R 4 R R " Z # (8.42) nO C hO C 1 gO C gO M dv C O C O ; 4 R R3 R where we have written R D 1 C gO C gO L2 .Mdv/ , for convenience. Then, according to Lemmas 5.14 and 7.11, we have now weakly convergent subsequences h h * in w-L1loc dtdxI w-L1 .M dv/ ; R 1 C jnj where h 2 L1loc .dtdxI L1 ..1 C jvj2 /M dv//, and hO hO * R 1 C jnj

in w-L2loc dtdxI w-L2 .M dv/ ;

where hO 2 L1loc .dtdxI L2 .M dv//. Hence, taking weak limits in (8.42), we find, further utilizing the strong convergence nO n ! R 1 C jnj in L2loc .dtdx/ from Lemma 7.11, that 2 3 1 jvj : h D hO C n u v C 2 2 2 Next, it is readily seen that the elementary decompositions q q 2 L gO ˙ D Q gO ˙ ; gO ˙ Q G˙ ; G˙ ; 2 q q ˙ 2 L gO ; gO D Q gO ˙ ; gO Q G˙ ; G ; 2

(8.43)


255

yield that

1 C C L hO D Q gO ; gO Q gO ; gO 2 q q p p 2 C C G ; G Q G ; G 2 Q (8.44)

1 C D Q gO gO nO ; gO C C Q gO ; gO C gO nO 2 Z

C 1 C C nO Q 1; gO C Q gO ; 1 qO qO M dv d; 3 2 2 R S and

1 C Q gO ; gO Q gO ; gO C L hO D 2 q q p p 2 C C G ; G Q G ; G 2 Q

1 C Q gO gO nO ; gO Q gO ; gO C gO nO D 2 Z

C; 1 qO qO;C M dv d: C nO Q 1; gO Q gO ; 1 2 R3 S2 (8.45) We also have the simple decomposition 0 0 1 C 1 gO gO gO C gO gO C gO gO C gO 2 2 C C 0 O0 D gO gO h gO gO hO C hO 0 nO hO nO : 2 2 2 2

qO C C qO qO C; qO;C D

As previously, we renormalize the above identities into " ! ! !# O hO hO C h D C Q gO ; L Q ; gO R 2 R R Z

C nO 1 C C Q 1; gO C Q gO ; 1 qO qO M dv d; 3 2 2R R " ! ! !# R S O O O h h h D Q ; gO Q gO ; L R 2 R R Z 1 C; nO C Q 1; gO Q gO ; 1 qO qO ;C M dv d; 2R R R3 S2

256


and 0 C 1 C qO C qO qO C; qO;C D gO gO R 2 !0 hO C nO 2 R

hO R

!0

hO C gO gO 2 R

!

hO nO : 2 R

Finally, taking weak limits, we find, utilizing again the strong convergence from Lemma 7.11, that Z C O L h D q q M dv d; R3 S2

1 1 L hO D nu ŒQ .1; v/ Q .v; 1/ C n Q 1; jvj2 Q jvj2 ; 1 2Z 4 C; ;C q q M dv d R3 S2 Z C; 1 1 q ;C M dv d; q D nu L.v/ C n L jvj2 2 4 R3 S2 and

q C C q q C; q ;C D 0;

which yields, in view of (8.43), that Z L .h/ D

R3 S2

C q q M dv d;

and 1 L .h/ D nu L.v/ C n L jvj2 2

Z R3 S2

q C; q ;C M dv d;

and concludes the proof of the proposition.

As a direct consequence of the previous propositions, we derive in the next result Ohm’s law and the internal electric energy constraint from (4.38). Proposition 8.5. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.7 for strong interspecies interactions, i.e., ı D 1. In accordance with Lemmas 5.1, 5.2, 5.13, 5.14 and 7.11 denote by g ˙ 2 L1 dtI L2 .M dxdv/ ; and

h 2 L1loc dtdxI L1 .1 C jvj2 /M dv ; E; B 2 L1 dtI L2 .dx/ ;


257

any joint limit points of the families gO ˙ and h defined by (5.3) and (5.29), E and B , respectively. Then, one has 1 C j nu D rx C E C u ^ B and w D n; 2 where ˙ , u and are, respectively, the densities, bulk velocity and temperature associated with the limiting fluctuations g ˙ , j and w are, respectively, the electric current and the internal electric energy associated with the limiting fluctuation h and the electric conductivity > 0 is defined by (2.70). Proof. By Proposition 8.3, we have that Z C Q q q C q C; q ;C ˆMM dvdv d R3 R3 S2 1 D rx .C / .E C u ^ B/ ; 2 Z C Q q q C q C; q ;C ‰MM dvdv d R3 R3 S2

D 0;

Q and ‰ Q are defined by (2.68). Then, where we have used the identity (2.69) and ˆ further incorporating identities (8.38) and (8.39) from Proposition 8.4 into the above relations yields that 2 Z jvj Q dv nu L.v/ C n L .L C L/ .h/ ˆM 2 R3 1 C rx . / .E C u ^ B/ ; D 2 2 Z jvj Q dv nu L.v/ C n L .L C L/ .h/ ‰M 2 R3 D 0: Finally, using (2.68) and the self-adjointness of L C L, we deduce that Z nu j D .nu ˆ C n ‰ h/ ˆM dv 3 ZR Q dv D .L C L/ .nu ˆ C n ‰ h/ ˆM R3 2 Z jvj Q dv nu L.v/ C n L D .L C L/ .h/ ˆM 2 R3 1 rx .C / .E C u ^ B/ ; D 2

258


3 .n w/ D 2

Z .nu ˆ C n ‰ h/ ‰M dv ZR

3

Q dv .L C L/ .nu ˆ C n ‰ h/ ‰M 2 Z jvj Q dv nu L.v/ C n L D .L C L/ .h/ ‰M 2 R3 D 0; D

R3

which concludes the proof of the proposition.

8.3 Energy inequalities for two species, strong interaction In view of the results from Section 8.2, we are now able to establish the limiting energy inequality for two species in the case of strong interactions. Proposition 8.6. Let f˙ ; E ; B be the sequence of renormalized solutions to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) considered in Theorem 4.7 for strong interspecies interactions, i.e., ı D 1. In accordance with Lemmas 5.1, 5.2, 5.3 and 7.11, denote by g ˙ 2 L1 dtI L2 .M dxdv/ ; h 2 L1loc dtdxI L1 1 C jvj2 M dv and q ˙ ; q ˙; 2 L2 MM dtdxdvdv d any joint limit points of the families gO ˙ , h , qO ˙ and qO˙; defined by (5.3), (5.29) and (5.6), respectively, and by E; B 2 L1 dtI L2 .dx/ any joint limit points of the families E and B , respectively. Then, one has the energy inequality, for almost every t 0, 1 1 knk2L2 C 2 kuk2L2 C 5 k k2L2 C kEk2L2 C kBk2L2 .t/ x x x x x 2 2 Z t 1 9 2 2 2 2 C 2 krx ukL2 C 5 krx kL2 C kj nukL2 C kw n kL2 .s/ ds x x x x 8

0 C in ; where ˙ , u and are, respectively, the densities, bulk velocity and temperature associated with the limiting fluctuations g ˙ and the charge density is given by n D C , while j and w are, respectively, the electric current and the internal electric energy associated with the limiting fluctuation h, and, finally, the viscosity > 0, thermal conductivity > 0, electric conductivity > 0 and energy conductivity

> 0 are respectively defined by (2.61), (2.70) and (2.71).

8.3 Energy inequalities for two species, strong interaction

259

Proof. First, by the estimate (5.7) from Lemma 5.3 and the weak sequential lower semi-continuity of convex functionals, we find that, for all t 0, Z Z Z ˙ 2 1 t MM dvdv d dxds q 4 0 R3 R3 R3 S2 Z Z Z ˙ 2 1 t qO MM dvdv d dxds lim inf !0 4 0 R3 R3 R3 S2 Z Z 1 t lim inf 4 D f˙ .s/ dxds; !0 R3 0 and

1 2

Z tZ

Z

˙; 2 MM dvdv d dxds q R3 R3 R3 S2 0 Z tZ Z ˙; 2 1 MM dvdv d dxds qO lim inf !0 2 0 R3 R3 R3 S2 Z Z 1 t D fC ; f .s/ dxds; lim inf 4 !0 R3 0

which, when combined with Lemma 5.1, yields, passing to the limit in the entropy inequality (4.36), for almost every t 0, Z Z 2 2 1 1 C 2 C g jEj C jBj2 .t/ dx g .t/M dxdv C 2 R3 R3 2 R3 Z Z Z 2 2 1 t 2 2 q C C q C q C; C q ;C MM dvdv d dxds C 4 0 R3 R3 R3 S2 in C : Since, according to Lemmas 5.10 and 5.11, the limiting fluctuations 2 3 jvj ˙ ˙ g D CuvC 2 2 are infinitesimal Maxwellians which differ only by their densities C and , we easily compute that, in view of the strong Boussinesq relation C C C D0 2 following from (8.33), 1 2

C 2 2

C . /2 3 C 2 C .g / M dv D C juj2 C 2 g 2 2 R3 1 2 5 D n C juj2 C 2 ; 4 2

Z

260


where n D C , which implies 5 1 1 1 2 2 2 2 2 knkL2 C kukL2 C k kL2 C kEkL2 C kBkL2 x x x x x 4 2 2 2 Z tZ Z C 2 1 q C q C; C q C q ;C MM dvdv d dxds C 16 0 R3 R3 R3 S2 Z Z Z 2 2 1 t q C q C q C; q ;C MM dvdv d dxds C 8 0 R3 R3 R3 S2 C in ; (8.46) where we have used the identity (8.40). There only remains to evaluate the contribution of the entropy dissipation in (8.46). To this end, applying the method of proof of Proposition 6.4, based on the Bessel inequality (6.16) (where and are now defined by (2.61) with ı D 1 instead of (2.29), which introduces a factor 2 in (6.16)), with the constraints (8.31) and (8.33) from Proposition 8.3, note that it holds Z t

2 krx uk2L2 C 5 krx k2L2 .s/ ds x x 0 Z tZ Z C 2 1 q C q C; C q C q ;C MM dvdv d dxds: 16 0 R3 R3 R3 S2 (8.47) Next, the remaining contributions in the entropy dissipation will be evaluated through a direct application of the following Bessel inequality: ˇ2 ˇZ ˇ C 8 ˇˇ C; ;C Q ˇ C q q q dvdv d q ˆMM ˇ ˇ R3 R3 S2 2 Z C 4 C; ;C Q (8.48) C q q q Cq ‰MM dvdv d

R3 R3 S2 Z 2 2 MM dvdv d; q C q C q C; q ;C R3 R3 S2

Q and ‰ Q are defined by (2.68). where ˆ For the sake of completeness, we provide a short justification of (8.48) below. But prior to this, let us conclude the proof of the present proposition. To this end, we employ the identities (8.38) and (8.39) from Proposition 8.4 in combination with the relations (2.68) and the self-adjointness of L C L to deduce from the inequality (8.48) that 9 8 jj nuj2 C .w n /2

Z (8.49) 2 2 q C q C q C; q ;C MM dvdv d: R3 R3 S2

Combining this with (8.46) and (8.47) concludes the proof of the proposition.

261

8.3 Energy inequalities for two species, strong interaction

Now, as announced above, we give a short proof of (8.48). To this end, for any vector A 2 R3 and any scalar a 2 R, one computes straightforwardly, employing the identities (2.69) and (2.71) and the collisional symmetries, that ˇ ˇ Z ˇ A ˆ Q Cˆ Qˆ Q0ˆ Q 0 C a ‰ Q0‰ Q 0 ˇ2 Q C‰ Q ‰ ˇ ˇ ˇ Q0 Q0 Q0 Q 0 ˇ MM dvdv d Q Q Q Q R3 R3 S2 A ˆ ˆ ˆ C ˆ C a ‰ ‰ ‰ C ‰ Z ˇ ˇ ˇ ˇ ˇA ˆ Q Cˆ Qˆ Q0ˆ Q 0 ˇ2 C ˇA ˆ Q ˆ Qˆ Q0Cˆ Q 0 ˇ2 D

R3 R3 S2

ˇ ˇ ˇ ˇ Q C‰ Q‰ Q0‰ Q 0 ˇ2 C ˇa ‰ Q ‰ Q‰ Q0C‰ Q 0 ˇ2 MM dvdv d C ˇa ‰ Z Z 2 Q Q M dv D 4 .A ˝ A/ W ˆ ˝ ˆ M dv C 4a ‰‰ R3

R3

D 2A A C 4 a2 : Therefore, defining, for any q0 ; q1 2 L2 .MM dvdv d /, the projection Q Cˆ Qˆ Q0ˆ Q 0 C a0 ‰ Q0‰ Q 0 Q C‰ Q ‰ A0 ˆ qN 0 ; D Q ˆ Q ˆ Q0Cˆ Q 0 C a0 ‰ Q0C‰ Q 0 Q ‰ Q‰ qN 1 A0 ˆ where A0 D

1 2 C

a0 D

Z R3 R3 S2

1 2 Z

1 4

Z

R3 R3 S2

R3 R3 S2

1 C 4

Q0ˆ Q 0 MM dvdv d Q Cˆ Qˆ q0 ˆ

Z

Q0Cˆ Q 0 MM dvdv d; Q ˆ Q ˆ q1 ˆ

Q0‰ Q 0 MM dvdv d Q C‰ Q ‰ q0 ‰

R3 R3 S2

Q0C‰ Q 0 MM dvdv d; Q ‰ Q ‰ q1 ‰

we find that Z q0 qN 0 MM dvdv d D 2A0 A0 C 4 a02 q q N1 3 3 2 1 R R S ˇ ˇ2 Z ˇ qN 0 ˇ ˇ ˇ D ˇ qN 1 ˇ MM dvdv d: R3 R3 S2 Hence, we obtain the Bessel inequality ˇ ˇ2 ˇ qN 0 ˇ ˇ ˇ D ˇ qN 1 ˇ MM dvdv d R3 R3 S2 ˇ ˇ 2 Z ˇ q0 ˇ ˇ ˇ ˇ q1 ˇ MM dvdv d: R3 R3 S2 Z

2A0 A0 C

4 a02

(8.50)

262


Therefore, setting q0 D q C q and q1 D q C; q ;C in (8.50), we find, exploiting the collisional symmetries of q ˙ and q ˙; , that ˇ2 ˇZ ˇ C 8 ˇˇ C; ;C Q q q q Cq ˆMM dvdv d ˇˇ ˇ R3 R3 S2 2 Z C 4 C; ;C Q C q q q Cq ‰MM dvdv d

R3 R3 S2 Z 2 2 q C q C q C; q ;C MM dvdv d; R3 R3 S2

which concludes the justification of (8.48).

Chapter 9

Approximate macroscopic equations The most difficult part of the asymptotic analysis consists in deriving the evolution equations for the bulk velocity and temperature insofar as they involve a singular limit and nonlinear advection terms. In particular, we expect the situation to be very different according to the asymptotic regime from Theorems 4.5, 4.6 and 4.7 under consideration. Indeed, the corresponding limiting systems, (4.31) and (4.39), respectively, do not enjoy the same stability properties: as explained in Chapter 3, the incompressible quasi-static Navier–Stokes–Fourier–Maxwell–Poisson system (4.31) is weakly stable in the energy space, which is not the case for the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with solenoidal Ohm’s law (4.39). Before focusing on this question of stability, we will first investigate the consistency of the electro-magneto-hydrodynamic approximation. For renormalized solutions (even though we cannot prove their existence, see Section 4.1), it is not known that conservation laws are satisfied. We therefore have to prove that approximate conservation laws hold and control their conservation defects. However, the uniform bounds established in Chapter 5 are not sufficient to do so: In the regime of Theorem 4.5 leading to the incompressible quasi-static Navier– Stokes–Fourier–Maxwell–Poisson system (4.31), we will also use the nonlinear weak compactness contained in Lemma 7.8. In the more singular regimes of Theorems 4.6 and 4.7 leading to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell systems with (solenoidal) Ohm’s laws (4.38) and (4.39), we are not able to establish such an a priori nonlinear control (compare Lemma 7.8 to the weaker nonlinear compactness statement of Lemma 7.10). The idea is therefore to use a modulated energy (or relative entropy) argument, in the same spirit as the weak-strong stability results of Chapter 3. In order to simplify the presentation, we will first detail the decompositions and convergence proof in the one-species case of Theorem 4.5, thus enlightening the points where the equi-integrability from Lemma 7.8 is required. We will then explain how to adapt these parts of the proof to the more singular regimes of Theorems 4.6 and 4.7.

264

9 Approximate macroscopic equations

9.1 Approximate conservation of mass, momentum and energy for one species In fact, in Chapter 8, we have already treated a similar singular limit (of lower order, though) in the regime of weak interactions for two species, which led to the derivation of the solenoidal Ohm’s law and internal electric energy constraint (see Proposition 8.2). Here, in order to deduce the limiting evolution equations for one species, we are confronted with an even more singular limit and face similar difficulties, which we briefly recall now. We have seen in Section 6.1 that it is possible to derive limiting kinetic equations of the type Z v rx g E v D qM dv d; R3 S2

from (4.28) (see (6.6) in the proof of Proposition 6.1). Here, we intend to take advantage of the symmetries of the collision integrand q to go one order further and, thus, to derive a singular limit. Of course, since we are considering renormalized fluctuations, we do not expect that the integrals in v of the right-hand side of the Vlasov– Boltzmann equation in (4.28) against collision invariants are zero, but they should converge to zero as ! 0 provided that we choose some appropriate renormalization which is sufficiently close to the identity. To estimate the ensuing conservation defects, we will also need to truncate large velocities. The precise construction is detailed below and will be essentially the same, later on in Section 9.2, for approximate conservation laws of mass, momentum and energy associated with (4.35). Note that, even if conservation laws were known to hold for renormalized solutions of (4.28) and (4.35), we would have to introduce similar truncations of large tails and large velocities in order to control uniformly the flux and acceleration terms (see [70]). Thus, similarly to the proof of Proposition 8.1, we start from the Vlasov–Boltzmann equation from (4.28) renormalized with the admissible nonlinearity .z/ defined by .z/ 1 D .z 1/.z/; 1 where 2 C .Œ0; 1/I R/ satisfies the following assumptions, for some given C > 0: .z/ 1; for all z 2 Œ0; 2; .z/ ! 0; as z ! 1; ˇ 0 ˇ C ˇ .z/ˇ ; for all z 2 Œ0; 1/: 3 .1 C z/ 2 Note that necessarily j.z/j With the notation

2C 1

.1 C z/ 2

D .G /; O D 0 .G /;

:

(9.1)

9.1 Approximate conservation of mass, momentum and energy. . .

265

the scaled Vlasov–Boltzmann equation in (4.28) renormalized relatively to the Maxwellian M with the admissible nonlinearity .z/ reads 1 1 @t .g / C v rx .g / C .E C v ^ B / rv .g / E vG O (9.2) 1 D 3 O Q .G ; G / : 2 We also introduce a truncation of large velocities jvj , with K D Kj log j, K for some large K > 0 to be fixed later on, and 2 Cc1 .Œ0; 1// a smooth compactly supported function such that 1Œ0;1 1Œ0;2 . 2 , where ' is Thus, multiplying each side of the above equation by '.v/ jvj K a collision invariant, and averaging with respect to Mdv leads to the approximate conservation laws 2 Z jvj @t M dv C rx F .'/ D A .'/ C D .'/; g ' (9.3) 3 K R with the notations 1 F .'/ D

Z

jvj2 g ' K R3

vM dv;

(9.4)

for the fluxes, 2 Z 1 jvj vM dv .1 C g /O ' A .'/ D E K R3 2 Z jvj M dv g .E C v ^ B / rv ' C 3 K R 2 2 Z Z 1 jvj jvj vM dv C E vM dv O ' g O ' D E K K R3 R3 2 Z jvj 2 0 jvj2 vM dv g ' C E 3 K K K R 2 Z jvj M dv; g .E C v ^ B / .rv '/ C K R3 (9.5) for the acceleration terms, and 2 Z 1 jvj D .'/ D 3 M dv; (9.6) O Q.G ; G /' R3 K for the corresponding conservation defects. By describing the asymptotic behavior of F .'/, A .'/ and D .'/, we will prove the following consistency result (compare with the formal macroscopic conservation laws (2.16)).

266


Proposition 9.1. Let .f ; E ; B / be the sequence of renormalized solutions to the scaled one-species Vlasov–Maxwell–Boltzmann system (4.28) considered in Theorem 4.5 and denote by Q , uQ and Q the density, bulk velocity and temperature associated 2 . with the renormalized fluctuations g jvj K Then, one has the approximate conservation laws 1 @t Q C rx uQ D R;1 ; Z juQ j2 Q qO MM dvdv d Id @t uQ C rx uQ ˝ uQ 3 R3 R3 S2

1 1 D rx Q C Q C E C Q E C uQ ^ B C R;2 ; Z 3Q 5 Q qO Q MM dvdv d @t Q C rx uQ 2 2 R3 R3 S2 D uQ E C R;3 ; Q where and Q are defined by (2.14) and (2.15), and the remainders R;i , i D 1; 2; 3,

1;1 converge to 0 in L1loc .dtI Wloc .dx//.

The proof of Proposition 9.1 consists in three steps respectively devoted to the study of conservation defects, fluxes and acceleration terms in (9.3). It does not present any particular difficulty and relies on refined decompositions of the different terms in the same spirit as the proof of Proposition 8.2. For the sake of clarity, these three steps are respectively detailed in Sections 9.1.1, 9.1.2 and 9.1.3, below. More precisely, Proposition 9.1 will clearly follow from the combination of the approximate conservation laws (9.3) with: Lemma 9.2, which handles the vanishing of conservation defects D .'/, for any collision invariant ', Lemma 9.3, which establishes the asymptotic behavior of the fluxes F .v/ and 2 F jvj2 52 , Lemma 9.4, which characterizes the acceleration terms A .1/, A .v/ and 2 A jvj2 52 as ! 0. In order to easily extend, later on in Section 9.2, the arguments from the present section to the case of two species, we are going to carefully keep track of and emphasize the different points where the equi-integrability property from Lemma 7.8 is used.

9.1.1 Conservation defects The first step of the proof is to establish the vanishing of conservation defects. Lemma 9.2. The conservation defects defined by (9.6) converge to zero. More precisely, for any collision invariant ', D .'/ ! 0 in L1loc .dtdx/ as ! 0:


267

Proof. Following the strategy of proof of Proposition 8.1, we introduce a convenient decomposition of D .'/, for any collision invariant ', and then estimate the different terms using the uniform bounds from Lemmas 5.2 and 5.3 (provided by the relative entropy and entropy dissipation), the relaxation estimate (5.11) from Lemma 5.10, as well as the equi-integrability coming from Lemma 7.8. Thus, using (5.8), we decompose D .'/, taking advantage of collisional symmetries: 2 Z jvj qO2 MM dvdv d O ' D .'/ D 4 R3 R3 S2 K 2 p Z 1 jvj qO G G MM dvdv d O ' 1 R3 R3 S2 K Z p 1 O .1 O / ' qO G G MM dvdv d C R3 R3 S2 (9.7) Z p 1 0 O O 1 O0 O ' qO G G MM dvdv d C R3 R3 S2 Z 0 O O O0 O ' qO 2 MM dvdv d 4 R3 R3 S2 def

D D1 .'/ C D2 .'/ C D3 .'/ C D4 .'/ C D5 .'/;

where we have used that ' is a collision invariant to symmetrize the last term. Now, we show that each term Di .'/, i D 1; : : : ; 5, vanishes separately. The vanishing of the first term D1 .'/, for any function '.v/ growing at most quadratically at infinity easily follows, using Lemma 5.3, from the estimate 2 1 2 D .'/ 1 kO kL1 jvj ' kqO k 2 1 L .dt dx/ L MM dt dxdvdv d 4 K L CK D CKj log j: The second term D2 .'/ is controlled by estimate (8.8) on the tails of Gaussian distributions. Using the bound from Lemma 5.3 and the pointwise boundedness of p 0 .z/ z, we get indeed, for all '.v/ growing at most quadratically at infinity, 2 D .'/ 1 L

loc .dt dx/

p C kqO kL2 MM dt dxdvdv d O G 1 L p 1fjvj2 K g ' G 2 2 Lloc dt dxIL .Mdv/

K

5

C 4 1 jlog j 4 ; which tends to zero as soon as K > 4.

L2 .Mdv/

268


The last term D5 .'/ is mastered using the same tools. For high energies, i.e., when jvj2 Kj log j, we obtain D5> .'/ Z def 0 D O O O0 O '1fjvj2 K g qO2 MM dvdv d 4 R3 R3 S2 1 p 2 ; O G 1 kO k2L1 kqO kL2 MM dvdv d '1fjvj2 K g 2 L L MM dvdv d so that, using the estimate (8.8) on the tails of Gaussian distributions and the bound on qO from Lemma 5.3,

K 5 ; D5> .'/ D O 4 1 jlog j 4 2 L .dt dx/

which tends to zero as soon as K > 4. For moderate energies, i.e., when jvj2 < Kj log j, we easily find Z def 5< 0 O O O0 O '1fjvj2 4.


297

p Moreover, since G˙ 2 implies gO ˙ 2. 2 1/, whence ˇ ˙ ˇ ˇˇ ˙ ˙2 ˇˇ ˇg ˇ D ˇgO C gO ˇ C gO ˙2 ; 4 we find, employing the uniform bounds from Lemmas 5.1 and 5.2, that Z Z ˇ ˙ ˇ ˙ ˇ 12 ˇ ˙ ˇ ˇ ˇg 1 ˙ ˇ .1 C jvj/ M dv C 12 ˇg ˇ ˇgO ˇ .1 C jvj/ M dv R3

R3

1 C g˙ 2 1 1 2

1Cjvj2

L

D o.1/L1dt IL1

Mdv

gO ˙

L2 .Mdv/

:

loc .dx/

Alternately, using Lemma 5.12, we obtain Z ˇ 2 ˇˇ ˇ ˙ ˇgO 1 ˙ C gO ˙2 1 ˙ ˇ .1 C jvj/ M dv C gO ˙ L2 ..1Cjvj/Mdv/ 4 R3 D o.1/L1 .dt dx/ : loc

If, instead of Lemma 5.12, one applies Lemma 9.9, then one finds that Z ˇ ˙2 ˇ 1 ˇ ˙ ˙ ˙ ˇ 2 ˇgO 1 C gO 1 ˇ 1 C jvj M dv 1 R3 4 Lloc .dx/ 2 n o ˙ C 1 G˙ 2 gO 2 Lloc dxIL2 .1Cjvj2 /Mdv 2 2 ! Z 1 ˙ 1 jvj ˙ ˙ C1 h g g M dxdv 2 3 3 2 K R R 2 2 ˙ ˙ jvj ˙ gN C o.1/L1 .dt / : C C2 g K 2 loc L .Mdxdv/ Thus, combining the preceding estimates with the decomposition (9.62), we arrive at Z R3

g˙ '.v/M

Z dv D R3

g˙ ˙

jvj2 K

'.v/M dv Co.1/L1dt IL1

; (9.63)

loc .dx/

'.v/ 2 L1 .dv/, and for all '.v/ such that 1Cjvj Z 2 1 jvj ˙ ˙ ˙ g '.v/M dv g 1 R3 K Lloc .dx/ 2 2 ! Z 1 ˙ 1 jvj M dxdv C1 h g g˙ ˙ 2 2 K R3 R3 2 2 ˙ ˙ jvj ˙ C C2 g gN C o.1/L1 .dt / ; loc K L2 .Mdxdv/

298


'.v/ 1 for all '.v/ such that 1Cjvj .dv/, which, when incorporated into (9.61), yields 2 2 L the approximate conservation law 0 0 11 2 Z a26 a35 C C jvj vM dv C E ^ B C @a34 a16 AA g C g @t @ K R3 a15 a24 Z 2 C C 1 1 jvj v ˝ vM dv C 2 m g C g C rx R3 K 2 jE j C jB j2 C Tr a rx .E ˝ E C e C B ˝ B C b / C rx 2 D @t o.1/L1dt IL1 .dx/ C RQ ; loc

where the remainder RQ satisfies 2 2 ! Z 1 C 1 jvj C C RQ 1;1 M dxdv C1 h g g Wloc .dx/ 2 2 K R3 R3 2 2 ! Z 1 1 jvj g C C1 h g M dxdv 2 2 K R3 R3 2 2 2 C C jvj jvj C gN ; g gN C C2 g K K L2 .Mdxdv/ C o.1/L1

loc .dt /

:

(9.64) Then, expressing the flux terms above with Lemma 9.12, we find that 0 11 0 a26 a35 @t @2uQ C E ^ B C @a34 a16 AA a15 a24 ! ˇ C ˇ2 ˇuQ ˇ C juQ j2 1 C rx uQ C QC Q Q Id C 2 m ˝u Cu ˝u 3 Z C Q rx qO C qO MM dvdv d R3 R3 S2 jE j2 C jB j2 C Tr a rx .E ˝ E C e C B ˝ B C b / C rx 2

2 D rx Q C Q C @t o.1/L1dt IL1 .dx/ C RN ; loc where the remainder RN also satisfies (9.64). Finally, an application of estimates (9.47) and (9.60) concludes the proof of the proposition.

299


9.2.5 Proofs of Lemmas 9.7, 9.8, 9.9 and 9.10 At last, we provide a complete justification of Lemmas 9.7, 9.8, 9.9 and 9.10. Proof of Lemma 9.7. This lemma hinges upon the simple fact that the renormalization .z/ enjoys the suitable bound from below (9.37). In terms of the renormalized fluctuations, this bound implies that ˇ ˇ 1 ˇˇ ˙ ˇˇ ˇˇ ˙ ˙ ˇˇ gO g C ˇgO ˙ ˇ ; C for some C > 1, which will be used repeatedly throughout the present proof. In orderpto establish the first bound (9.44), notice that, since G˙ 2 implies gO ˙ 2. 2 1/, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ1n ˙ o gO ˙ ˇ C ˇ1n ˙ o ˙ g ˙ ˇ ˇ G 2 ˇ ˇ G 2 ˇ ˇ ˇ ˇ ˙ ˙ ˇ ˇ ˇ ˙ˇ ˙ˇ o n ˇ ˇ C g gN C C ˇ1 G ˙ 2 gN ˇ ˇ ˇ ˇ ˇˇ ˇ N ˙ ˇ C C ˇgO ˙ ˇ ˇgN ˙ ˇ ; C ˇg˙ ˙ g whence 2 1n ˙ o gO ˙ jvj C g˙ ˙ N ˙ L2 .Mdv/ g G 2 K L2 .Mdv/ ˙ gN ˙ C CK gO L2 .Mdv/ : 1 C jvj2 L1 .dt dxdv/ Moreover, it is readily seen that 2 C 1n ˙ o gO ˙ 1 jvj 1 gO ˙ L2 .1Cjvj2 /Mdv : G 2 K L2 .Mdv/ K2 Therefore, it follows that, combining the preceding estimates and considering the uniform bounds from Lemmas 5.2 and 5.12, 1n ˙ o gO ˙ C g˙ ˙ N ˙ L2 .Mdv/ g G 2 2 L .Mdv/ ! 1 C O .j log j/L1 dt IL2 .dx/ C O ; 1 j log j 2 L2 .dt dx/ loc

which concludes the proof of (9.44).

300


To deduce the second bound (9.45), we decompose, writing g˙ D gO ˙ 1 C 4 gO ˙ and using that 1 C 4 gO ˙ ˙ is uniformly bounded pointwise, ˇ ˇ ˇ ˇ 2 2

ˇ ˇ ˙ ˙ ˇ ˇ ˇg jvj gO ˙ ˇ D ˇgO ˙ 1 C gO ˙ ˙ jvj 1 ˇ ˇ ˇ ˇ ˇ K 4 K ˇ ˇ 2

ˇ ˙ ˙ ˇ ˙ ˙ jvj ˇ C ˇg 1 ˇˇ 1 C gO 4 K ˇ ˙ ˙ ˇ ˙ˇ ˇ C g gN ˇ ˇ 2 ˇ ˙ ˇ ˙ ˙ jvj ˇ C C ˇgN 1 C gO 1 ˇˇ 4 K ˇ 2 ˇ ˇ ˙ ˙ ˇ ˇ ˇ jvj ˇ C ˇg gN ˙ ˇ C C ˇˇgN ˙ 1 ˇ K ˇ 2 ˇ ˇ jvj ˇˇ C C ˇˇgN ˙ gO ˙ ˙ K ˇ ˇ 2 ˇ ˇ jvj ˇˇ C C ˇˇgN ˙ ˙ 1 ; K ˇ n o p which implies, further using that ˙ 1 is supported on gO ˙ 2. 2 1/ , 2 ˙ ˙ jvj ˙ g gO 2 K L .Mdv/ ˙ ˙ gN ˙ ˙ ˙ C g gN L2 .Mdv/ C CK gO L2 .Mdv/ 1 C jvj2 1 L .dt dxdv/ 2 ˙ g N jvj 2 CC : 2 .1 C jvj / 1 K 2 L .Mdv/ 1 C jvj L1 .dt dxdv/ Then, employing the control of Gaussian tails (8.8) and the uniform bound from Lemma 5.2, we infer that 2 ˙ ˙ g jvj gO ˙ K L2 .Mdv/ ˙ ˙ K 5 C g gN ˙ L2 .Mdv/ C O .j log j/L1 dt IL2 .dx/ C C 4 j log j 4 ; which establishes (9.45). The third bound (9.46) easily follows from the estimate ˙ ˙ g gO ˙ 2 L .Mdv/ 2 2 ˙ ˙ ˙ jvj jvj ˙ gO g C gO 1 2 K K L2 .Mdv/ L .Mdv/ 2 ˙ ˙ C jvj gO ˙ C 1 gO ˙ L2 1Cjvj2 Mdv ; g K 2 L .Mdv/ K2 which, when combined with the second bound (9.45), concludes its justification.


301

The fourth bound (9.47) is simple. It is just a matter of noticing, by virtue of p ˙ ˙ ˙ assumptions (9.1) and (9.36), and writing 2g D gO 1 C G , that ˙ ˙ ˙ g ˙ o n g 2 C 1 p L .Mdv/ 1 C G˙ G˙ 2 2 L .Mdv/ C D ; gO ˙ 1n ˙ o 2 G 2 L2 .Mdv/ and then using estimate (9.44) to conclude. The justification of (9.48) is simple, as well. Since …gN ˙ D gN ˙ , we easily estimate ˙ ˙ O gN ˙ 2 gO …gO ˙ 2 C C …gN ˙ …gO ˙ L2 .Mdv/ L .Mdv/ C g L .Mdv/ C gO ˙ gN ˙ L2 .Mdv/ : Therefore, the bound (9.48) is obtained by combining the preceding control with (9.46) and (9.47). We focus now on (9.49). We first easily find that 2 ˙ 2 ˙ jvj …gO 1 q K L .Mdv/ 2 ˙ ˙ 2 ˙ jvj 2 ˙ 1 C … gO gN C C gN q L2q .Mdv/ K L .Mdv/ ˙ ˙ 2 C gO gN L2 .Mdv/ 2 gN ˙ 2 jvj ˙ 4 1 CC 1 C jvj q 1 C jvj2 L1 .dt dxdv/ K L .Mdv/ 2 gN ˙ 2 1 C jvj4 jvj 1 CC : q 1 C jvj2 1 K L .Mdv/ L .dt dxdv/ Therefore, utilizing the control of Gaussian tails (8.8) and the fact that G˙ 2 on the support of ˙ 1, we deduce that 2 ˙ 2 ˙ jvj …gO 1 q K L .Mdv/ ˙ 2 2 1 K 2 C gO gN ˙ L2 .Mdv/ C CK2 q gO ˙ Lq 2 .Mdv/ C C j log j2C 2q 2q ; which, when combined with (9.46) and (9.47), concludes the proof of (9.49).

302


Next, we establish the last bound (9.50). Note first that the case p p D 2 is easily ˙ ˙ deduced from (9.44), using again that G 2 implies gO 2. 2 1/. Thus, the difficulty here lies in obtaining a gain of velocity integrability. To this end, we introduce the macroscopic truncation ˙ D 1

g˙ ˙ gN ˙

L2 .M dv/

1

:

Then, we have ˙ ˙ 1 n o g gN ˙ 2 C : 1 1 G ˙ 2 p L .Mdv/ L .Mdv/ Moreover, controlling Gaussian tails with (8.8), it clearly holds that ˙

2 1 1 K 1n ˙ o 1 jvj C 2p 1 jlog j 2p ; G 2 K Lp .Mdv/

which is small as soon as K 8 > 2p, so that we only have to control the size of 2 1 n jvj o 1 G ˙ 2 K on the support of ˙ . Thus, employing the decomposition ˇ ˇ 3 ˇ ˇ ˇ ˇ 1 n 3 1 G ˙ 2o 1nG ˙ 2o ˇgO ˙ ˇ 1nG ˙ 2o ˇ…gO ˙ ˇ C ˇgO ˙ …gO ˙ ˇ ; 2 2 we find, for any 1 < r < 2, the interpolation estimate 2 1 n jvj o 1 G ˙ 2 2 K 2 ˇ ˇ ˇ 2 ˇ ˇ C n jvj ˇˇ ˙ o … gO gN ˙ ˇ C ˇ…gN ˙ ˇ C ˇgO ˙ …gO ˙ ˇ r 1 ˙ 2 2 G K 2 r ˇ ˙ ˇ2 r2 ˇ ˙ ˇ 2 ˇ… gO gN ˙ ˇ r C 1n ˙ o ˇgO ˇ G 2 2 r

C CK

4 r 2

2 ˇ ˙ ˇ 2r gN ˙ r ˇgO ˇ 1 C jvj2 1 L

C

C .dt dxdv/

2 r2

ˇ ˙ ˇ2 ˇgO …gO ˙ ˇ r :


303

Therefore, combining the preceding estimate with the relaxation (5.11) from Lemma 5.10, we deduce that 2 1 1n ˙ o jvj 2 G 2 K Lr .Mdv/ 2 2 r ˙ 2 ˙ o n … gO gN ˙ r 2 C 1 G˙ 2 gO 2 L 2r .Mdv/ L .Mdv/

2r ˙ 2r C gO ˙ 2 2 gO 2 C C O./ 2 L .dt dx/ L .Mdv/ L .Mdv/ 2 2 r 2 2 o ˙ n C C C gO ˙ gN ˙ L2 .Mdv/ 1 G˙ 2 gO 2 L .Mdv/ 4 2 4r 2 4 4 2 r 2 C C jlog j r r gO ˙ Lr 2 .Mdv/ C C r 2 g˙ ˙ L2 .Mdv/ C O Lr .dt dx/ 2 2 o ˙ n C C C gO ˙ gN ˙ L2 .Mdv/ 1 G ˙ 2 gO 2 r

C C jlog j

4 r 2

L2 .Mdv/

2 4 4 4 C C jlog j r 2 gO ˙ Lr 2 .Mdv/ C C r 2 g˙ ˙ N ˙ Lr 2 .Mdv/ g 4 4 4 C C r 2 gN ˙ Lr 2 .Mdv/ C O r 2 r : 2 r

L .dt dx/

Note that these controls do not yield vanishing remainders in the endpoint case r D 2, which explains the use of the interpolation parameter 1 < r < 2. Then, recalling that the preceding estimate only needs to be performed on the support of ˙ and denoting p D 2r, we infer that 2 jvj ˙ 1 n o 1 G ˙ 2 K Lp .Mdv/ ˙ n o C 1 G ˙ 2 gO C C gO ˙ gN ˙ L2 .Mdv/ C C g˙ ˙ N ˙ L2 .Mdv/ g L2 .Mdv/ 4 2 4 2 4 4 C C jlog j p p 1 gO ˙ Lp2 .Mdv/ C C p 1 gN ˙ Lp2 .Mdv/ C O p 1 p ; L .dt dx/

which, when combined with the bounds (9.44), (9.46) and (9.47), concludes the proof of (9.50). The proof of the lemma is now complete. Proof of Lemma 9.8. This lemma is a simple consequence of the relaxation estimate provided by Lemma 5.10: gO ˙ …gO ˙ D O./L1 dt dxIL2 .Mdv/ : loc

304


In orderpto establish the first bound (9.51), notice that, since G˙ 2 implies gO ˙ 2. 2 1/, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ1n ˙ o gO ˙ ˇ ˇgO ˙ …gO ˙ ˇ C ˇ1n ˙ o …gO ˙ ˇ ˇ ˇ G 2 ˇ G 2 ˇ ˇ ˇ ˇ ˙ ˇ ˇ ˇ ˙ ˙ 2 n o ˇgO …gO ˇ C C ˇˇ1 G ˙ 2 1 C jvj ˇˇ gO L2 .Mdv/ ;

whence 2 2 1n ˙ o gO ˙ jvj gO ˙ …gO ˙ L2 .Mdv/ C CK gO ˙ L2 .Mdv/ : G 2 K L2 .Mdv/ Moreover, it is readily seen that 2 1n ˙ o gO ˙ 1 jvj G 2 K

L2 .Mdv/

C gO ˙ L2 .1Cjvj2 /Mdv : 1 K2

Therefore, combining the preceding estimates and considering the uniform bounds from Lemmas 5.2 and 5.12, we have 1n ˙ o gO ˙ G 2

L2 .Mdv/

D o.1/L1

loc .dt dx/

;

which concludes the proof of (9.51) by interpolation. To deduce the second bound (9.52), we decompose, writing g˙ D gO ˙ 1 C 4 gO ˙ and using that 1 C 4 gO ˙ ˙ is uniformly bounded pointwise, ˇ ˇ 2 ˇ ˇ ˙ ˙ ˇg jvj gO ˙ ˇ ˇ ˇ K ˇ ˇ 2 ˇ ˙ ˇ ˙ ˙ jvj ˇ D ˇgO 1 C gO 1 ˇˇ 4 K ˇ ˇ 2 ˇ ˙ ˇ ˇ ˇ jvj C ˇgO …gO ˙ ˇ C C ˇˇ…gO ˙ 1 C gO ˙ ˙ 1 ˇˇ 4 K ˇ 2 ˇ ˇ ˇ ˇ ˇ jvj ˇ C ˇgO ˙ …gO ˙ ˇ C C ˇˇ…gO ˙ 1 ˇ K ˇ ˇ 2 ˇ 2 ˇ ˇ ˇ ˙ ˙ ˇ jvj ˇˇ ˇ…gO 1 jvj ˇ ; C C ˇˇ…gO ˙ gO ˙ ˙ C C ˇ ˇ K K ˇ

305


…gO which implies, since 1Cjvj 2

L1 .Mdv/

C kgO kL2 .Mdv/ , that

2 ˙ ˙ g jvj gO ˙ K L2 .Mdv/ ˙ 2 ˙ C gO …gO L2 .Mdv/ C CK gO ˙ L2 .Mdv/ 2 ˙ jvj 2 gO 2 C C .1 C jvj / 1 L .Mdv/ : K L2 .Mdv/ Then, employing the control of Gaussian tails (8.8) and the uniform bound from Lemma 5.2, we infer that 2 ˙ ˙ ˙ gO …gO ˙ 2 g jvj gO ˙ C 2 L .Mdv/ K L .Mdv/ C O .j log j/ L1 dt IL1 .dx/

K 5 C O 4 j log j 4

;

L1 dt IL2 .dx/

which, with an interpolation argument, establishes (9.52). The third bound (9.53) easily follows from the estimate ˙ ˙ g gO ˙ 2 L .Mdv/ 2 2 ˙ ˙ ˙ jvj jvj ˙ gO g C gO 1 2 K K L2 .Mdv/ L .Mdv/ 2 ˙ ˙ C jvj gO ˙ C 1 gO ˙ L2 1Cjvj2 Mdv ; g K 2 L .Mdv/ K2 which, when combined with the second bound (9.52), concludes its justification. Next, simply notice that the fourth bound (9.54) is a reformulation of Lemma 5.10 with an interpolation argument, which we have incorporated here for mere convenience. Finally, we easily establish the last bound (9.55). To this end, note first that the ˙ ˙ case p p D 2 is easily deduced from (9.51), using again that G 2 implies gO 2. 2 1/. Furthermore, repeating the estimate leading to the bound (9.11) yields here that, for every 2 p < 4, 1 n 1 ˙ o D O.1/L2 dt dxILp .Mdv/ : loc G 2 Therefore, the bound (9.55) is obtained by interpolation. The proof of the lemma is thus complete.

306


Proof of Lemma 9.9. First, we utilize (5.25), with some fixed > 4 therein to be determined later on, to estimate 1 q ˇ ˇ ˇ ˙ ˇ 2 12 .1Cjvj/2 ˇ ˇ 2 2 2 ˇ ˙ ˇ2 ˙ .1 C jvj/ gO h g .1 C jvj/ ˇgO ˇ C e 2 .1 C jvj/ ˇgO ˙ ˇ ˇ ˇ ˇ ˇ2 2 12 .1Cjvj/2 1 2 2 h g˙ C .1 C jvj/2 ˇgO ˙ ˇ C e 2 .1 C jvj/ ˇgO ˙ ˇ ; 2 whence 1 ˇ ˇ ˇ ˇ 4 ˙ 4 2 .1Cjvj/2 2 ˇ ˙ ˇ2 e 2 .1 C jvj/ ˇgO ˙ ˇ : .1 C jvj/ gO 2 h g C It follows that, for any 2 < p < 4, 2 1n ˙ o gO ˙ 2 G 2 L .1Cjvj/2 Mdv Z 4 1nG ˙ 2o h g˙ M dv 2 R3 1 2 2 2 .1Cjvj/2 ˙ n n o o 2 C2 1 g O C 2 e .1 C jvj/ 1 G˙ 2 2 G˙ 2 2 L .Mdv/ L .Mdv/ Z 2 4 o ˙ n 2 1n ˙ o h g˙ M dv C 2 1 G˙ 2 gO 2 R3 G 2 L .Mdv/ 2 2 1 .1Cjvj/2 n o e 2 .1 C jvj/ C 2 : 2p 1 G˙ 2 p p2 L .Mdv/ L .Mdv/ (9.65) p Here we need to set the parameter so large that p2 < 2 in order to yield a finite constant in the last term above. Then, further using that h.z/ D 12 .z.1 C z//2 C O.z 3 /, we deduce that 2 ˙ o n 1 ˙ gO G 2 2 2 L

.1Cjvj/ Mdv

˙ 2 ˙ ˙ 2 h g M dv g G˙ 2 2 R3 2 2 1 ˙ og o n n C C2 O C C 1 1 2 ˙ G˙ 2 2 G 2 Lp .Mdv/ L .Mdv/ Z 1 ˙ 1 ˙ ˙ 2 M dv C1 h g g 2 2 R3 2 2 1 ˙ n n o o C C2 1 G ˙ 2 gO C C2 1 G ˙ 2 p L2 .Mdv/ L .Mdv/ ˇ ˇ Z ˇ ˙ 2 ˙ ˙ 2 ˇ C2 ˇ M dv C 2 1n ˙ o ˇh g g R3 G 0 denote diverse constants which only depend on fixed parameters and which we do not distinguish for simplicity. Finally, combining the preceding estimate with the bounds (9.44), (9.47) and (9.50), modulating the last term jg˙ ˙ j2 D jg˙ ˙ gN ˙ j2 C 2 g˙ ˙ gN ˙ gN ˙ C jgN ˙ j2 , and using that ˇ ˇ 1nˇˇˇg ˙ ˇˇˇ 0 so large that h.z/ < 12 .z.1 C z//2 implies jzj N , for any z 2 Œ1; 1/, which is always possible in view of the assumptions (9.1) on .z/. Then using that h.z/ D 12 .z.1 C z//2 C O.z 3 / again, we infer that 2 1n ˙ o gO ˙ G 2

2 2 L2 loc dxIL .1Cjvj/ Mdv

2 2 ! 1 ˙ 1 jvj ˙ ˙ g C1 h g M dxdv 2 2 K R3 R3 2 1 ˙ o o n n C C2 1 G ˙ 2 C C2 1 G ˙ 2 gO 2 2 L2 Lloc dxILp .Mdv/ loc dxIL .Mdv/ ˇ 2 ˇ2 Z ˇ ˙ˇ ˇ ˙ ˙ jvj ˇˇ ˇ nˇ o ˇ ˇ ˇ C C2 1 ˇˇg ˙ ˇˇN g ˇg M dxdv: K ˇ R3 R3 Z

Then, as before, combining the preceding estimate with the bounds (9.44), (9.47) and (9.50), and modulating the last term ˇ2 ˇ 2 ˇ2 ˇ 2 ˇ ˙ ˙ ˇ ˇ2 ˇ jvj ˇˇ jvj 1 ˇˇ ˙ ˙ ˙ˇ ˇ gN ˇ C ˇgN ˙ ˇ ; ˇg g ˇ ˇ 2 K K we arrive at 2 1n ˙ o gO ˙ G 2

2 2 L2 loc dxIL .1Cjvj/ Mdv

2 2 ! 1 ˙ 1 jvj M dxdv h g C1 g˙ ˙ 2 2 K R3 R3 2 2 ˙ ˙ jvj ˙ C C2 g gN C o.1/L1 .dt / loc K L2 .Mdxdv/ Z ˇ ˇ ˇ ˇ2 C C2 1nˇˇˇg ˙ ˇˇˇN o ˇg˙ ˇ ˇgN ˙ ˇ M dxdv: Z

R3 R3


309

Finally, since gN ˙ belongs to L1 .dtI L2 .Mdxdv// and 1fjg ˙ j 0 and thermal conductivity > 0 are defined by (2.29). In particular, it holds that .; u; ; B/ 2 L1 Œ0; 1/; dtI L2 R3 ; dx ; .u; / 2 L2 Œ0; 1/; dtI HP 1 R3 ; dx : This energy bound can be improved to the actual energy inequality (3.2) provided some well-preparedness of the initial data is assumed. This is discussed in the few remarks following the statement of Theorem 4.5.

11.1.3 Evolution equations We move on now to the rigorous derivation of the asymptotic macroscopic evolution equations. We know from Chapter 9 that some approximate macroscopic evolution equations, which look like the Navier–Stokes–Fourier system with electromagnetic forces, are satisfied up to a remainder which is small in some distribution space. More precisely, according to Proposition 9.1, defining the macroscopic variables Q , uQ and Q as the density, bulk velocity and temperature, respectively, corresponding to the 2 used therein, it holds that renormalized fluctuations g jvj K 8 1 ˆ ˆ @t Q C rx uQ D R;1 ; ˆ ˆ ˆ ˆ Z ˆ ˆ juQ j2 ˆ ˆ Q qO MM Id @t uQ C rx uQ ˝ uQ ˆ dvdv d ˆ ˆ 3 R3 R3 S2
0 are defined in (2.29) and , are the kinetic fluxes defined by (2.14). Incorporating the above relations into (11.7) and recalling that u is a solenoidal vector field, we finally find the evolution system @t .u C A/ C rx .u ˝ u/ x u D rx p C E C u ^ B; 5 3 5 C rx u x D u E: @t 2 2 2 Then, defining the adjusted electric field by EQ D P EQ C P ? EQ D @t A C rx . C / ; the above evolutions system, when combined with the constraint equations (11.2) and (11.3), can be recast as 8 ˆ @t u C u rx u x u D rx p C EQ C rx C u ^ B; ˆ ˆ ˆ ˆ div u D 0; ˆ ˆ ˆ < 3 5 3

x . C / D ; @t C u rx x D 0; ˆ 2 2 2 ˆ ˆ ˆ ˆ rot B D u; div EQ D ; ˆ ˆ ˆ : div B D 0; @t B C rot EQ D 0; which is precisely the incompressible quasi-static Navier–Stokes–Fourier–Maxwell– Poisson system (4.31).

11.1.4 Temporal continuity, initial data and conclusion of the proof There only remains to establish the weak temporal continuity of the observables: .; u; ; B/ 2 C Œ0; 1/I w-L2 R3 ; dx ; (11.8) and to identify their respective initial data.

11.1 Proof of Theorem 4.5

329

For the moment, we only know from the weak formulation (11.6) that, for any solenoidal '.x/ 2 Cc1 .R3 I R3 / and .x/ 2 Cc1 .R3 I R/, Z .u C A/ .t; x/'.x/ dx 2 C.Œ0; 1/I R/; R3 Z 3 .t; x/.x/ dx 2 C.Œ0; 1/I R/; 2 R3 in .u C A/ .0; x/ D uin 0 C A0 .x/; 3 3 in .0; x/ D 0 0in .x/: 2 2 Notice, replacing the test function ' by rot ', that one also has Z .rot u C B/ .t; x/'.x/ dx 2 C.Œ0; 1/I R/; and

R3

for any '.x/ 2 Cc1 .R3 I R3 /, and

in .rot u C B/ .0; x/ D rot uin 0 C B0 .x/:

In particular, a straightforward density argument yields that 3 rot u C B; 2 C Œ0; 1/I w-L2 R3 ; dx : 2 Finally, using the relations rot B D u and x . C / D , it is easy to express each observable , u, and B in terms of rot u C B and 32 , only: 2 x 3 x . C / 5 x 3 D ; D 3 5 x 3 5 x 2 rot rot B x u D .rot u C B/ ; uD 1 x 1 x 1 x 2 2 x 3 D D ;

x 3 5 x 2 1 rot uD .rot u C B/ : BD x 1 x It follows that (11.8) holds true and that the initial data are provided by

x 3 5 x 1 x .t D 0/ D 3 5 x .t D 0/ D

in 30 20in ;

rot 1 x in 1 30 20in ; B.t D 0/ D 1 x u.t D 0/ D

which, at last, concludes the proof of Theorem 4.5.

in rot uin 0 C B0 ; in rot uin 0 C B0 ;

Chapter 12

The renormalized relative entropy method We are now going to investigate the more singular asymptotics leading to the twofluid incompressible Navier–Stokes–Fourier–Maxwell systems with (solenoidal) Ohm’s law. As explained in Section 3.2, the limiting models obtained in these regimes are not weakly stable and, thus, are not known to have global solutions (except under suitable regularity and smallness assumptions on the initial data). However, from the physical point of view, these asymptotic regimes are important insofar as they justify Ohm’s laws, which are fundamental in plasma physics. From the mathematical point of view, the Navier–Stokes–Fourier–Maxwell systems obtained in the limit share many features with the three-dimensional incompressible Euler equations. Proving some convergence results requires then methods which are different from the weak compactness techniques used in the proof of Theorem 4.5 in Chapter 11 and which are typically based on weak-strong stability principles and dissipative solutions (see Section 3.2.3). The main novelty here, compared to the convergence results from the Boltzmann equation to the incompressible Euler equations (see [70, Chapter 5]), is to use renormalization techniques together with the relative entropy method.

12.1 The relative entropy method: old and new The principle of the relative entropy method is to compare the distribution with its formal asymptotics in some appropriate metrics: The idea of using the relative entropy H.f˙ jM / to build such metrics goes back to Yau [78] in the framework of the asymptotic study of Ginzburg– Landau’s equation, then to Golse [14, Section 2.3.5] for the hydrodynamic limits of the Boltzmann equation. The important points of this method are that the scaled relative entropy is a Lyapunov functional for the Boltzmann equation and that it controls the size of the fluctuations. An approximate solution is obtained by formal expansions (the so-called Hilbert expansions; see [42]), which consist in seeking a formal solution to the scaled system in the form ˙ fapp D M 1 C g0˙ C 2 g1˙ C : Note that the successive approximations gn˙ should depend a priori both on macroscopic variables t, x and on fast variables t , t2 , . . . , x , x2 , . . . For well-prepared initial data, that is for data satisfying some profile condition (thermodynamic equilibrium) as well as macroscopic linear constraints

332

12 The renormalized relative entropy method

(incompressibility and Boussinesq relations, for instance), there are neither a kinetic initial layer nor fast oscillating waves, so that g0˙ reduces actually to the solution of the limiting system. Note that, in the cases considered here, the nonlinear constraints (Ohm’s laws) have a different status: solutions of the limiting models are well-defined even though these constraints are not satisfied initially. This is similar to the existence theory for parabolic equations with initial data which are not in the domain of the diffusion operator. The core of the proof consists then in getting some stability inequality for the scaled modulated entropy ! Z X 1 X 1 f˙ ˙ ˙ ˙ ˙ ˙ f log ˙ f C fapp dxdv; H f jfapp D 2 2 R3 R3 fapp ˙

˙

which measures in some sense the distance between the fluctuations g˙ and their expected limits g0˙ . The convergence relies then on some technical computations and Grönwall’s lemma. The stability inequality we expect to obtain should be reminiscent of the inequality defining the corresponding dissipative solutions of the limiting systems (see Section 3.2.3). Thus, it should be based solely on the decay of the entropy and on local conservation laws. In particular, there is no need for a priori strong compactness: nonlinear terms should be controlled by a loop estimate using Grönwall’s lemma. Unfortunately, this simple strategy fails, in general: even for weak solutions in the sense of distributions (not renormalized), provided they exist, we have no control on large velocities; for renormalized solutions in the sense of DiPerna and Lions, provided they exist, local conservation laws are not known to hold. The main novelty here is to use renormalization techniques combined with the relative entropy method. More precisely, we will not use the usual modulated entropy inequality for renormalized solutions to the kinetic equations. Rather, we will modulate a renormalized version of the entropy inequality, which requires much less a priori information on the solutions.

12.2 Proof of Theorem 4.6 on weak interactions Several steps of this demonstration have been performed in the preceding chapters. We therefore begin our proof by appropriately gathering previous results together.

12.2.1 Weak convergence of fluctuations, collision integrands and electromagnetic fields Thus, we are considering here a family of renormalized solutions .f˙ ; E ; B / to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35), in the regime of

12.2 Proof of Theorem 4.6 on weak interactions

333

weak interspecies interactions, i.e., ı D o.1/ and ı is unbounded, satisfying the scaled entropy inequality (4.36). By Lemmas 5.1 and 5.2, the corresponding families of fluctuations g˙ and renormalized fluctuations gO ˙ are weakly compact in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and L2loc .dtI L2 .Mdxdv//, respectively, while, by Lemma 5.3, the corresponding collision integrands qO ˙ and qO ˙; are weakly compact in L2 .MM dtdxdvdv d /. Thus, using Lemma 5.1 again and (5.5), we know that there exist g ˙ 2 L1 .dtI L2 .Mdxdv//, .E; B/ 2 L1 .dtI L2 .dx// and q ˙ ; q ˙; 2 L2 .MM dtdxdvdv d /, such that, up to extraction of subsequences, in L1loc dtdxI L1 1 C jvj2 Mdv ; g˙ * g ˙ in L1 dtI L2 .Mdxdv/ ; gO ˙ * g ˙ (12.1) .E ; B / * .E; B/ in L1 dtI L2 .dx/ ; in L2 MM dtdxdvdv d ; qO˙ * q ˙ qO˙; * q ˙; in L2 MM dtdxdvdv d ; as ! 0. Therefore, one also has the weak convergence of the densities ˙ , bulk ˙ ˙ velocities u˙ and temperatures corresponding to g : ˙ * ˙ ;

u˙ * u˙

and ˙ * ˙

in L1loc .dtdx/ as ! 0;

where ˙ ; u˙ ; ˙ 2 L1 .dtI L2 .dx// are, respectively, the densities, bulk velocities and temperatures corresponding to g ˙ . In fact, Lemma 5.10 implies that 2 3 jvj : g ˙ D …g ˙ D ˙ C u˙ v C ˙ 2 2 Next, we further introduce the scaled fluctuations h D

ı C g g n ;

where n D C is the charge density, and the electrodynamic variables j D

ı C u u ;

w D

ı C ;

which are precisely the bulk velocity and temperature associated with the scaled fluctuations h . In view of Lemma 5.13, the electric current j and the internal electric energy w are uniformly bounded in L1loc .dtdx/, which necessarily implies, letting ! 0, that uC D u and C D . Furthermore, Proposition 6.2 asserts that ı C D , as well. Therefore, we appropriately rename the limiting macroscopic variables D C D ; u D uC D u ; D C D ;

334


and the limiting fluctuation gDg

C

Dg DCuvC

jvj2 3 : 2 2

Now, by Lemma 5.14, h is weakly compact in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and j and w are weakly compact in L1loc .dtdx/, so that, up to extraction of subsequences, there are h 2 L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and j; w 2 L1loc .dtdx/ such that h * h in L1loc dtdxI L1 1 C jvj2 Mdv ; j * j

in L1loc .dtdx/;

w * w in L1loc .dtdx/; as ! 0. Moreover, by Proposition 6.3, one has the infinitesimal Maxwellian form 2 3 jvj : h D j vCw 2 2

12.2.2 Constraint equations, Maxwell’s system and the energy inequality In view of Proposition 6.2, we already know that the limiting thermodynamic fields , u and satisfy the incompressibility and Boussinesq relations div u D 0;

C D 0:

(12.2)

Moreover, the discussion in Section 6.4 shows that the limiting electromagnetic field satisfies the following form of Maxwell’s equations: 8 @t E rot B D j; ˆ ˆ ˆ < @ B C rot E D 0; t ˆ div E D 0; ˆ ˆ : div B D 0: Note that, taking the divergence of the Ampère equation above, one necessarily has div j D 0. Finally, Proposition 8.2 further establishes that the electrodynamic variables j and w satisfy the solenoidal Ohm’s law and the internal electric energy equilibrium relation j D .rx pN C E C u ^ B/ ; w D 0; where the electric conductivity > 0 is defined by (2.75) and the pressure gradient rx pN is the Lagrange multiplier associated with the solenoidal constraint div j D 0. As for the energy bound, Proposition 6.5 states that, for almost every t 0,

1 2 kuk2L2 C 5 k k2L2 C kEk2L2 C kBk2L2 .t/ x x x x 2 Z t 1 2 krx uk2L2 C 5 krx k2L2 C kj k2L2 .s/ ds C in ; C x x x 0


335

where the viscosity > 0, thermal conductivity > 0 and electric conductivity > 0 are respectively defined by (2.61) and (2.75). In particular, it holds that .u; ; E; B/ 2 L1 Œ0; 1/; dtI L2 R3 ; dx ; .u; / 2 L2 Œ0; 1/; dtI HP 1 R3 ; dx ; j 2 L2 Œ0; 1/ R3 ; dtdx : This energy bound can be improved to the actual energy inequality

1 2 kuk2L2 C 5 k k2L2 C kEk2L2 C kBk2L2 .t/ x x x x 2 Z t 1 2 krx uk2L2 C 5 krx k2L2 C kj k2L2 .s/ ds C x x x 0

1 in 2 2 2 2 2 u L2 C 5 in L2 C E in L2 C B in L2 ; x x x x 2 using the well-preparedness of the initial data (4.41).

12.2.3 The renormalized modulated entropy inequality We move on now to the rigorous derivation of a stability inequality encoding the asymptotic macroscopic evolution equations for u and in the spirit of the weakstrong stability inequalities used in Section 3.2.3 to define dissipative solutions for some Navier–Stokes–Maxwell systems. Recall that, as explained therein, such systems are in general not known to display weak stability, so that their weak solutions in the energy space are not known to exist. 2 As in Section 9.2, we define the renormalized fluctuations g˙ ˙ jvj , with K 1 K D Kj log j, for some large K > 0, and 2 Cc .Œ0; 1// a smooth compactly supported function such that 1Œ0;1 1Œ0;2 , and where ˙ D .G˙ / for some renormalization 2 C 1 .Œ0; 1/I R/ satisfying (9.1). We also consider here an auxiliary renormalization 2 C 1 .Œ0; 1/I R/ satisfying (9.36), which will be used later on when applying estimates from Section 9.2, and denote ˙ D .G˙ /. jvj2 ˙ Since, up to further extraction of subsequences, K converges almost everywhere towards 1, g˙ is weakly compact in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and g˙ ˙ is uniformly bounded in L1 .dtI L2 .Mdxdv//, we deduce, by the Product Limit Theorem, that 2 jvj g˙ ˙ * g in L1 dtI L2 .Mdxdv/ : K We obtain similarly that g˙ ˙

jvj2 K

* g

in L1 dtI L2 .Mdxdv/ :

336


Therefore, one has the weak convergence of the densities Q˙ , bulk velocities uQ ˙ and jvj2 ˙ ˙ ˙ temperatures Q corresponding to g K :

Q˙ * ;

in L1 dtI L2 .dx/ as ! 0:

Q˙ *

uQ ˙ * u and

In particular, the hydrodynamic variables Q D C Q CQ 2

C

Q CQ , 2

uQ D

C

u Q Cu Q 2

and Q D

also obviously verify

Q * ;

uQ * u and Q *

in L1 dtI L2 .dx/ as ! 0:

(12.3)

It follows that, since u is solenoidal,

P ? uQ * 0

in L1 dtI L2 .dx/ as ! 0;

and, in view of the limiting Boussinesq relation, Q C Q * 0 in L1 dtI L2 .dx/ as ! 0:

(12.4)

(12.5)

Q We establish now the convergence of the electric current jQ D ı uQ C u . p Since gO ˙ 2 2 1 on the support of 1 ˙ , we easily estimate, using the uniform bound from Lemma 5.12, that ˇ ˇ ˇ ˇ ˇı ˙ ˇ ˇ ı ˙ ˙2 ˇ ˙ ˙ ˇ g 1 ˇ D ˇ gO C gO 1 ˇˇ ˇ ˇ ˇ 4 ˇ ˇ ˇı ˙ ˇ ı ˙2 ˙ ˇ ˇ ˇ gO 1 ˇ C gO 4 ˙2 ; C ıgO D O.ı/

1 L1 loc dt dxIL

1Cjvj2 Mdv

whereas, using the Gaussian decay (8.8), we also obtain, provided K > 4, that ˇ 2 ˇ 2 ˇ ˇ ˇı ˙ ˇ ˇ ı ˙ ˙2 ˇ jvj jvj ˇ g ˇ D ˇ gO C gO ˇ 1 1 ˇ ˇ ˇ ˇ K 4 K ˇ 2 ˇ ˇı ˇ ı ˙2 jvj ˇ C gO ˇˇ gO ˙ 1 ˇ 4 K 2 2 ı jvj C 2 1 C C ıgO ˙2 K D O.ı/L1 dt dxIL1 1Cjvj2 Mdv : loc

Thus, we infer that ı ˙ uQ u˙ !0


(12.6)

337


whence

jQ * j

in L1loc .dtdx/ as ! 0:

2 Now, the L2 .Mdxdv/ norm of g˙ ˙ jvj is not a Lyapunov functional but K it is nevertheless controlled by the relative entropy 2 2 C 1 g ˙ ˙ jvj 2 H f˙ ; (12.7) 2 K L2 .Mdxdv/ for some C > 1, and therefore by the initial data (4.42). One may therefore try, in a preliminary attempt to show an asymptotic inequality, to modulate the stability

jvj2 ˙ ˙ approximate energy associated with g K , i.e., its L2 .Mdxdv/ norm, by introducing a test function gN in infinitesimal Maxwellian form: 2 5 jvj N ; gN D uN v C 2 2 where

N x/ 2 C 1 Œ0; 1/ R3 with div uN D 0; u.t; N x/; .t; c

and then establishing a stability inequality for the modulated energies 2 2 1 g ˙ ˙ jvj gN : 2 2 K L .Mdxdv/

(12.8)

2 Notice that, by the elementary identity a2 C 32 b 2 D 35 .a C b/2 C 52 3b2a , for 5 any a; b 2 R, we have 2 2 ˙ ˙ jvj g gN 2 K L .Mdxdv/ 2 2 jvj ˙ ˙ gN … g 2 K L .Mdxdv/ 2 2 ˙ 2 ˙ 3 D Q C N L2 .dx/ C uQ uN L2 .dx/ C Q˙ N 2 L .dx/ 2 0 1 2 3Q ˙ 2Q˙ 2 3 5 2 A: D @ Q˙ C Q˙ 2 C uQ ˙ N L2 .dx/ C N u 2 L .dx/ 5 2 5 L .dx/

It turns out that this approach is not quite suitable for our purpose because, even though, for any 0 t1 < t2 (see the proof of Lemma 5.1), Z t2 Z t2 1 1 ˙ H f dt; (12.9) kgk2L2 .Mdxdv/ dt lim inf 2 !0 t1 2 t1

338


it is not possible to set C D 1 in (12.7). Indeed, the first term in the polynomial expansion of the function h.z/ D .1 C z/ log.1 C z/ z defining the entropy is 12 z 2 , but the second term is 16 z 3 and may be negative. Some entropy (or energy) is therefore lost by considering the modulated energies (12.8). These considerations lead us to introduce a more precise modulated functional in replacement of (12.8), capturing more information on the fluctuations. To be precise, instead of (12.8), we consider now the renormalized modulated entropies 2 Z 1 1 ˙ jvj ˙ ˙ N 2L2 .Mdxdv/ : (12.10) gM N dxdv C kgk H f g 2 3 3 K 2 R R Note that the above functional may be negative for fixed > 0. However, in view of (12.9), it recovers asymptotically a non-negative quantity, which is precisely the asymptotic modulated energy: Z t2 2 1 5 N 2L2 .dx/ C N L2 .dx/ dt ku uk 2 t1 2 Z t2 1 N 2L2 .Mdxdv/ dt D kg gk t1 2 (12.11) 2 Z Z t2 1 ˙ jvj gM N dxdv H f g˙ ˙ lim inf !0 2 K t1 R3 R3 1 N 2L2 .Mdxdv/ dt; C kgk 2 for all 0 t1 < t2 . The first term in (12.10) is precisely the entropy of f˙ and will be controlled by the scaled entropy inequality (4.36), whereas the last term in (12.10) only involves smooth quantities and will therefore be controlled directly. As for the middle term in the modulated entropy (12.10), its time derivative 2 will involve the approximate ˙ ˙ macroscopic conservation laws for g jvj . K Recall that a major difficulty in the relative entropy methods developed for the hydrodynamic limit of the Boltzmann equation towards the incompressible Euler equations (see [69, 70, 71]) pertains to the handling of large velocities. Here, large velocities are no longer a problem,

we deal now with conservations laws of renormal 2 for jvj ˙ ˙ ized fluctuations g K whose defects are well controlled. Thus, the present method is more robust than the usual relative entropy method which cannot deal with fluctuations of temperature. Furthermore, thanks to the flatness assumption (9.1) on .z/ near z D 1, the conservation defects are expected to vanish in the limit (at least formally). In fact, employing results from Section 9.2, they will be estimated in terms of the modulated entropy and entropy dissipation. The convergence will then be obtained through a loop estimate based on an appropriate use of Grönwall’s lemma.


339

Now, in order to establish the renormalized modulated entropy inequality leading to the convergence stated in Theorem 4.6, we introduce further test functions N x/; B.t; N x/; jN .t; x/ 2 Cc1 Œ0; 1/ R3 with div EN D div BN D div jN D 0; E.t; and we define the renormalized modulated entropy ıH .t/ D

1 C 1 H f C 2 H f 2 2 Z C C jvj gM N dxdv C kgk N 2L2 .Mdxdv/ g C g K R3 R3 2 2 1 1 C E EN L2 .dx/ C B BN L2 .dx/ 2Z 2 1 1 C Tr m C Tr a dx 2 R3 2 0 0 11 Z a26 a35 @ E EN ^ B BN C @a34 a16 AA uN dx; R3 a15 a24

where the matrix measures m and a are the defects introduced in Section 4.1.4 and controlled by the scaled entropy inequality (4.36). We also define the renormalized modulated energy 2 2 2 2 1 jvj 1 jvj C C gN gN ıE .t/ D g C g 2 2 K 2 K L2 .Mdxdv/ L .Mdxdv/ Z 2 2 1 1 1 1 C E EN L2 .dx/ C B BN L2 .dx/ C Tr m C Tr a dx 2 2 2 R3 2 0 0 11 Z a26 a35 @ E EN ^ B BN C @a34 a16 AA uN dx; R3 a15 a24 which is asymptotically equivalent to ıH .t/, at least formally. Note that ıH .t/ controls more accurately the large values of the fluctuations g˙ than ıE .t/. It is to be emphasized that ıH .t/ is voluntarily built with a renormalization .z/ merely satisfying (9.1), whereas ıE .t/ uses a nonlinearity .z/ verifying the more stringent hypotheses (9.36). Anyway, Lemma 12.1 below shows how the modulated entropy ıH .t/ controls the modulated energy ıE .t/. Finally, we introduce the renormalized modulated entropy dissipation ıD .t/ D

1 C 1 kqO qN k2 qO C qN C 2 2 MM dxdvdv d L L2 MM dxdvdv d 4 4 2 1 C qO C; qN C; L2 MM dxdvdv d 4 1 2 C qO ;C qN ;C L2 MM dxdvdv d ; 4

340


where

qN ˙ D rx uN W Q C Q Q 0 Q0 C rx N Q C Q Q 0 Q 0 ; (12.12) 1 qN ˙; D jN v v v 0 C v0 ; 2 so that Z qN ˙ M dv d D rx uN W L Q C rx N L Q D rx uN W C rx N ; R3 S2 Z 1 qN ˙; M dv d D jN L .v/ ; 2 R3 S2 with , , Q and Q defined by (2.14) and (2.15). Then, assuming from now on that kuk N L1 .dt dx/ < 1 and using the lower weak sequential semi-continuity of the entropies (12.11) and of the electromagnetic energy (3.25) together with Lemma 4.4, we find that, for all 0 t1 < t2 , Z t2 Z t2 Z t2 ıE .t/ dt lim inf min ıH .t/ dt; ıE .t/ dt ; (12.13) 0 !0

t1

t1

t1

where 2 2 1 1 ıE .t/ D kg gk N 2L2 .Mdxdv/ C E EN L2 .dx/ C B BN L2 .dx/ 2 2 Z E EN ^ B BN uN dx R3

2 5 1 N 2 2 C E EN L2 .dx/ L .dx/ 2 2 Z 2 1 C B BN L2 .dx/ E EN ^ B BN uN dx; 2 R3

D ku uk N 2L2 .dx/ C

while, repeating mutatis mutandis the computations leading to (6.20) and (6.22) in the proof of Proposition 6.5, we obtain, for all 0 t1 < t2 , Z t2 Z t2 ıD .t/ dt lim inf ıD .t/ dt; (12.14) t1

!0

t1

where 2 2 1 ıD .t/ D 2 krx .u u/k N 2L2 C 5 rx N L2 C j jN L2 x x x 1 1 2 q C qN C L2 MM dxdvdv d C kq qN k2 2 L MM dxdvdv d 4 4 2 1 C q C; qN C; L2 MM dxdvdv d 4 1 2 C q ;C qN ;C L2 MM dxdvdv d : 4


341

The following lemma shows how the modulated entropy ıH controls the modulated energy ıE up to a small remainder. Lemma 12.1. It holds that ıE .t/ C ıH .t/ C o.1/L1.dt / ; for some fixed constant C > 1. Proof. Recall first, in view of the hypotheses on the renormalizations (9.1) and (9.36), that ˇ ˙ ˙ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇg ˇ C ˇgO ˙ ˇ and 1 ˇgO ˙ ˇ ˇg ˙ ˙ ˇ C ˇgO ˙ ˇ ; C for some C > 1, and that the elementary inequality (B.5) implies 1 ˙2 1 gO 2 h g˙ : 4 We proceed then with the straightforward estimate, where C0 > 1 is a large constant and 0 < 2 is a small parameter to be determined later on: 2 2 1 jvj ˙ ˙ gN 1nˇˇˇgO ˙ ˇˇˇ>o g 2 K 2 1 2 nˇ C0 ˙ jvj ˙ ˙ g N C 1 ˇˇgO ˙ ˇˇˇ>o h g g g N 2 K 2 2 1 1 jvj gN C gN 2 1nˇˇˇgO ˙ ˇˇˇ>o C0 2 h g˙ g˙ ˙ K 2 2 jvj g1 N nˇˇˇgO ˙ ˇˇˇ>o C g˙ C0 ˙ ˙ K 2 1 2 nˇ CK jlog j ˙2 1 ˙ jvj ˙ ˙ gN C gN 1 ˇˇgO ˙ ˇˇˇ>o C C0 2 h g g gO : K 2

(12.15) Next, utilizing the simple inequality (B.6) and the fact that .z/

.z/

1 on ˇ ˚ ˇp ˇ z 1ˇ , we deduce that 2 2 2 1 jvj ˙ ˙ g gN 1nˇˇˇgO ˙ ˇˇˇo 2 K ! 2 2 2 1 2 1 jvj jvj ˙ ˙ ˙ ˙ g gN C gN 1nˇˇˇgO ˙ ˇˇˇo D g 2 K K 2 2 1 1 ˙ jvj gN C gN 2 1nˇˇˇgO ˙ ˇˇˇo h g g˙ ˙ 2 K 2 2 3 jvj C 1nˇˇˇgO ˙ ˇˇˇo : g˙ ˙ 6 K

342


Then, we find that 2 2 1 jvj g N 1nˇˇˇgO ˙ ˇˇˇo g˙ ˙ 2 K 2 1 2 nˇ 1 ˙ jvj ˙ ˙ gN C gN 1 ˇˇgO ˙ ˇˇˇo h g g 2 K 2 2 2 ˇ ˇ 1 jvj ˇ o ˇ nˇ g˙ ˙ gN ˇg˙ ˙ C 1 ˇˇ gO ˙ ˇˇ 6 K 2 ˇ ˇ 1 1 2 ˇˇ ˙ ˙ ˇˇ nˇ jvj ˇ o ˇ nˇ gN ˇg˙ ˙ C g˙ ˙ N g 1 ˇˇgO ˙ ˇˇˇo 1 ˇˇ gO ˙ ˇˇ g 3 K 6 2 1 1 ˙ jvj gN C gN 2 1nˇˇˇgO ˙ ˇˇˇo h g g˙ ˙ 2 K 2 2 2 jvj gN 1nˇˇˇgO ˙ ˇˇˇo C CK jlog j gO ˙2 ; C C1 g˙ ˙ K whence, setting sufficiently small so that the left-hand side remains positive,

2 2 1 jvj ˙ ˙ gN 1nˇˇˇgO ˙ ˇˇˇo g C1

2 K 2 1 2 nˇ 1 ˙ jvj ˙ ˙ g N C 1 ˇˇgO ˙ ˇˇˇo h g g g N 2 K 2

(12.16)

C CK jlog j gO ˙2 : Therefore, combining estimates (12.15) and (12.16) and using the uniform bound on gO ˙ from Lemma 5.2, we obtain that 2 2 1 jvj 1 ˙ ˙ g gN ; 1 2C1

min C0 2 K 2 1 2 1 ˙ jvj ˙ ˙ g N g N C h g g 2 K 2 C O . jlog j/L1 dt IL1 .Mdxdv/ ;

which, upon integrating against Mdxdv and adding the contributions of the electromagnetic field and the defect measures to the energy, concludes the proof of the lemma. The following result establishes the renormalized modulated entropy inequality at the order , which will eventually allow us to deduce the crucial weak-strong stability of the limiting thermodynamic fields, thus defining dissipative solutions.


343

Proposition 12.2. One has the stability inequality Z Rt 1 t ıH .t/ C ıD .s/e s . /d ds 2 0 ıH .0/e

Rt 0

.s/ds

0

uQ uN 3 Q 5 N Q 2

2

B C B C B R C C; ;C A B 2 R3 R3 S2 qO qO vMM dvdv d jNC .s/dx B C 3 R @ A N B E EN C uN ^ B N N B B C E E ^ uN

Z tZ C 0

e

Rt s

1

. /d

ds C o.1/L1 ; loc .dt /

(12.17) where the acceleration operator is defined by 1 0 1 02 .@ uN C P .uN r u/ N ^ BN N u/ N C P j A1 t x x C 2 @t N C uN rx N xN A2 C B B C B B C 1 N B C; N N N N N N A u; N ; j ; E; B D BA3 C D B j C P E C uN ^ B C @A A @ A @tEN rot BN CjN 4 A5 @t BN C rot EN and the growth rate is given by

.t/ D C

N N N ku.t/k W 1;1 .dx/ C k@t u.t/k L1 .dx/ C j .t/ L1 .dx/ 1 ku.t/k N L1 .dx/

! 2 C N .t/W 1;1 .dx/ ; W 1;1 .dx/

N C .t/

with a constant C > 0 independent of test functions and . Proof. The main ingredients of the proof of this stability inequality are: The scaled entropy inequality (4.36): Z Z 1 1 1 1 C 2 2 H f H f Tr m dx C j C jB j C Tr a C C jE dx 2 2 2 2 R3 2 R3 Z tZ 2 2 1 2 2 qO C C qO C qO C; C qO ;C MM dxdvdv d ds C 4 0 R3 R3 R3 S2 Z in 2 1 in 1 1 Cin C 2 H f C jE j C jBin j2 dx; 2 H f 2 R3 (12.18)

344


which is naturally satisfied by renormalized solutions of the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) (provided they exist), and where we have used the inequality (5.7) from Lemma 5.3 in order to conveniently simplify the dissipation terms. The approximate conservation of energy obtained in Proposition 9.5: Z qO C C qO Q 3Q 5 Q @t Q C rx uQ MM dvdv d 2 2 2 R3 R3 S2 D R;3 ; (12.19) where the remainder R;3 satisfies 1 kR;3 kW 1;1 .dx/ C ıE .t/ C C ıE .t/ıD .t/ 2 C o.1/L1 .dt / ; (12.20) loc

loc

for some C > 0. Note that we do not employ the approximate conservation of momentum from Proposition 9.5, which is crucial (see comments following the present proof, below). The approximate conservation of momentum law from Proposition 9.6: 0 0 11 1 1 a26 a35 @t @uQ C E ^ B C @a34 a16 AA 2 2 a a 15

24

2

C rx

1 juQ j Id C 2 m 3 2 ! C qO C qO Q MM dvdv d 2 R3 R3 S2

uQ ˝ uQ Z

(12.21)

1 rx .E ˝ E C e C B ˝ B C b / 2 jE j2 C jB j2 C Tr a C rx 4

1 D rx Q C Q C @t .R;8 / C R;7 ; where the remainders R;7 and R;8 satisfy ; R;8 D o.1/ L1 dt IL1 loc .dx/

kR;7 kW 1;1 .dx/ C1 ıH .t/ C C2 ıE .t/ C o.1/L1 loc

; loc .dt /

(12.22)

for some C1 ; C2 > 0. The approximate Ohm’s law Z ˙ qO ˙; vMM dvdv d D rx pN .E C uQ ^ B /CR;9 CR;10 ; R3 R3 S2

(12.23)


345

where the remainder R;9 vanishes weakly R;9 D o.1/w-L1

loc .dt dx/

;

(12.24)

whereas R;10 satisfies kR;10 kL1 .dx/

C ıE .t/ C o.1/w-L1 .dt / : loc 1 kuk N L1 .dx/

(12.25)

This approximate law is obtained directly from the limiting laws derived in Proposition 8.1. Indeed, it is easily deduced from (8.2) that (12.23) holds with the remainders Z ˙; R;9 D .E E/ ˙ qO q ˙; vMM dvdv d R3 R3 S2

R;10

C .uQ u/ ^ B C u ^ .B B/ ; O .ıE .t/ C ıE .t//L1 .dx/ D .uQ u/ ^ .B B/ D ; 1 kuk N L1 .dx/

where we have used that

1 kuk N L1 .dx/ B BN 2 2 E EN 2 2 C L .dx/ L .dx/ 2

2 2 1 E EN 2 C B BN L2 .dx/ L .dx/ 2Z E EN ^ B BN uN dx:

(12.26)

R3

The above estimate on R;10 is then readily improved to (12.25) upon noticing from (12.13) that ıE .t/ ıE0 .t/; where ıE0 .t/ is the limit, up to extraction of subsequences, of ıE .t/ in w L1 .dt/, and then writing ıE .t/ C ıE .t/ 2ıE .t/ C ıE0 .t/ ıE .t/ D 2ıE .t/ C o.1/w-L1 .dt / : It is to be emphasized that it would be possible to derive the above approximate Ohm’s law directly from Proposition 9.5. However, the method used here is more robust and we find it much more satisfiying to derive the consistency of the approximate law from the knowledge of the limiting equation. Indeed, morally, it is much more difficult to derive limiting equations, which require some kind of weak stability property, than to merely establish the consistency of approximate equations.

346


Maxwell’s equations 8 @t E rot B ˆ ˆ ˆ < @ B C rot E t ˆ div E ˆ ˆ : div B

D j D jQ C R;11 ; D 0; D ın ; D 0;

(12.27)

where the remainder R;11 D jQ j satisfies, thanks to the convergence (12.6), R;11 D o.1/L1

loc .dt dx/

:

(12.28)

Notice that we cannot rigorously write the identities (1.10) and (1.12) for the above system, because the source terms j and n do not belong to L2loc .dtdx/ a priori. Nevertheless, one has the following modulated identities: @t E EN C B BN C rx E ^ BN C EN ^ B (12.29) D jQ R;11 EN jN C A4 E A5 B ; and @t

ˇ ˇ2 ˇ ˇ2 1 E EN ^ B BN C rx ˇE EN ˇ C ˇB BN ˇ 2 N rx E E ˝ E EN C B BN ˝ B BN

1 D @t .E ^ B / C rx jE j2 C jB j2 rx .E ˝ E C B ˝ B / 2 C jN C A4 ^ B BN C E EN ^ A5 N C jQ R;11 ^ BN C ın E: (12.30)

The asymptotic characterization (6.7) of the limiting collision integrands q ˙ from Proposition 6.2, which implies that Z qO C C qO Q MM dvdv d rx uQ C rxt uQ ! 0; 2 R3 R3 S2 Z 5 qOC C qO Q MM dvdv d rx Q ! 0; 2 2 R3 R3 S2 (12.31) in the sense of distributions, where ; > 0 are defined by (2.61). The asymptotic characterization (6.11) of the limiting collision integrands q ˙; from Proposition 6.3, which implies that ! Z qO C; qO;C (12.32) vMM dvdv d C jQ ! 0; 2 R3 R3 S2

347


in the sense of distributions, where > 0 is defined by (2.75). Moreover, since j is solenoidal, it holds that ! Z qO C; qO ;C ? P vMM dvdv d * 0 in L2 .dtdx/: 3 3 2 2 R R S (12.33) Now, by the definition of the acceleration operator A, straightforward energy computations, similar in the proof of Proposition 3.3, applied to the to those performed N jN ; E; N BN , show that the following identities holds: test functions u; N ; Z d N 2L2 C jN ^ BN uN dx; kuk N 2L2 2 krx uk x x dt R3 R3 Z d 2 A2 N dx D kN k2L2 2 rx N L2 ; x x 3 dt Z Z R 1 A3 jN dx D kjN k2L2 C EN jN jN ^ BN uN dx; x R3 R3 Z

Z d 1 N 2 2 C kBk N 22 kEk A4 EN C A5 BN dx D EN jN dx: Lx Lx 2 dt R3 R3 Z

A1 uN dx D

All in all, combining the preceding expressions yields the energy identity: 0 1 uN B 5 N C Z B2 C d N NC E .t/ C DN .t/ D AB B j C dx; dt R3 @ EN A BN

(12.34)

where the energy EN and energy dissipation DN are defined by 1 2 1 2 EN .t/ D kgk N 2L2 .Mdxdv/ C EN L2 .dx/ C BN L2 .dx/ 2 2 2 2 5 1 1 2 2 D kuk N L2 .dx/ C N L2 .dx/ C EN L2 .dx/ C BN L2 .dx/ ; 2 2 2 and 2 1 2 DN .t/ D 2 krx uk N 2L2 C 5 rx N L2 C jNL2 x x x 1 1 2 2 D qN C L2 MM dxdvdv d C kqN k 2 L MM dxdvdv d 4 4 2 2 1 1 C qN C; L2 MM dxdvdv d C qN ;C L2 MM dxdvdv d : 4 4

348


Next, similar duality computations applied to the approximate conservation of energy (12.19) yield that Z Z d 3Q 3Q 3Q N N N uN rx dx Q dx C Q x Q dt R3 2 2 2 R3 Z Z qO C C qO Q 5 Q uQ MM dvdv d rx N dx 2 2 R3 R3 R3 S2 Z 1 3Q Q D R;3 N A2 dx: 2 2 R3 Further reorganizing the preceding equation so that all remainder terms appear on its right-hand side, we find that Z d 3Q Q N dx dt R3 2 Z Z C Q C qO C qO MM dvdv d rx N dx R3 R3 R3 S2 Z

5 1 3Q Q N N N Q C .uQ u/ D R;3 A2 N rx dx 2 2 2 R3 Z

5 ? C P uQ rx N 2 C uN rx N x N Q C Q dx 4 R3 Z Z qO C C qO Q 5 Q N N C x C MM dvdv d rx dx: 2 2 R3 R3 R3 S2 It then follows, using the convergences (12.4), (12.5), (12.31), the estimate (12.20) and Lemma 12.1 (allowing to control the energy by the entropy), that Z d 3Q Q N dx dt R3 2 Z Z C C qO C qO Q MM dvdv d rx N dx R3 R3 R3 S2

1 C N W 1;1 .dx/ ıE .t/ C .ıE .t/ıD .t// 2 Z (12.35) 1 3Q Q dx C o.1/w-L1 .dt / A2 loc 2 R3 2 2 1 C N W 1;1 .dx/ C N W 1;1 .dx/ ıH .t/ ıD .t/ 4 Z 1 3Q Q dx C o.1/w-L1 .dt / : A2 loc 2 R3 2


349

Likewise, using the solenoidal property div uN D 0, analogous duality computations applied to the approximate conservation of momentum (12.21) yield that 1 0 1 0 Z a26 a35 d 1 1 @uQ uN C .E ^ B / uN C @a34 a16 A uN R;8 uN A dx dt R3 2 2 a a 15 24 1 0 1 0 Z a a 26 35 @ 1 .E ^ B / @t uN C 1 @a34 a16 A @t uN R;8 @t uN A dx 3 2 2 a a R 15 24 Z 1 .P uQ / ˝ uN uQ ˝ uQ 2 m W rx uN dx C 3 2 R Z Z qO C C qO Q C MM dvdv d W rx uN x uN uQ dx 2 R3 R3 R3 S2 Z 1 .E ˝ E C e C B ˝ B C b / W rx uN dx C 2 R3 Z 1 N 1 D R;7 uN C P j ^ BN uQ A1 uQ dx; 2 2 R3 whence, reorganizing some terms so that remainders are moved to the right-hand side, 0 0 1 1 Z a26 a35 1 1 d @uQ uN C .E ^ B / uN C @a34 a16 A uN A dx dt R3 2 2 a a 15 24 Z Z C Q C qO C qO MM dvdv d W rx uN dx R3 R3 R3 S2 Z 1 N 1 N R;7 uN C P j ^ B uQ A1 uQ dx D 2 2 R3 Z ? ? 1 C uN ˝ P uQ C P uQ ˝ uN C .uQ u/ N ˝ .uQ u/ N C 2 m W rx uN dx 2 R3 Z Z C qO C qO Q uQ x uN C C MM dvdv d W rx uN dx 2 R3 R3 R3 S2 0 0 1 1 Z a a 26 35 @ 1 .E ^ B / @t uN C 1 @a34 a16 A @t uN R;8 @t uN A dx C 3 2 2 a a R 15 24 Z Z d 1 .E ˝ E C e C B ˝ B C b / W rx uN dx C R;8 uN dx: 2 R3 dt R3

350


Then, using the convergences (12.4), (12.31), the estimates (4.20), (4.25), (12.22) and Lemmas 4.4 and 12.1 (allowing to control the energy by the entropy), we arrive at 0 1 1 0 Z a26 a35 d 1 1 @uQ uN C .E ^ B / uN C @a34 a16 A uN A dx dt R3 2 2 a a 15 24 Z Z C Q qO C qO MM dvdv d W rx uN dx C R3 R3 R3 S2 ! Z N L1 .dx/ k@t uk 1 ıH .t/ C kuk N W 1;1 .dx/ C A1 uQ dx 1 kuk N L1 .dx/ 2 R3 Z 1 d P jN ^ BN uQ dx C o.1/w-L1 .dt / C o.1/L1.dt / C loc dt 2 R3 Z 1 ..E ^ B / @t uN .E ˝ E C e C B ˝ B C b / W rx u/ N dx: C 2 R3 The next step consists in combining the preceding inequality with the identity (12.30) in order to modulate the Poynting vector E ^ B . This yields 0 0 1 1 Z a26 a35 1 1 d @uQ uN C E EN ^ B BN uN C @a34 a16 A uN A dx dt R3 2 2 a a 15 24 Z Z C Q W rx uN dx C qO C qO MM dvdv d R3 R3 R3 S2 ! N L1 .dx/ k@t uk N W 1;1 .dx/ C ıH .t/ C kuk 1 kuk N L1 .dx/ Z Z 1 1 A1 uQ dx C A4 ^ B BN C E EN ^ A5 uN dx 2 R3 2 R3 d C o.1/w-L1 .dt / C o.1/L1.dt / loc dt Z 1 ın EN R;6 ^ BN uN .uQ ^ B / jN uN ^ BN jQ C 2 R3 jN ^ BN P ? uQ dx Z 1 C jN ^ B BN .uN uQ / C E EN ^ B BN @t uN dx 2 R3 Z 1 E EN ˝ E EN C e C B BN ˝ B BN C b W rx uN dx: 2 R3


351

It then follows, using the convergence (12.4), the estimates (4.25), (12.28) and Lemmas 4.4 and 12.1 (allowing to control the energy by the entropy), that 0 1 1 0 Z a26 a35 d 1 1 @uQ uN C E EN ^ B BN uN C @a34 a16 A uN A dx dt R3 2 2 a a 15 24 Z Z C Q qO C qO MM W rx uN dx C dvdv d R3 R3 R3 S2 N W 1;1 .dx/ C k@t uk N L1 .dx/ C jNL1 .dx/ kuk C ıH .t/ 1 kuk N L1 .dx/ Z Z 1 1 A1 uQ dx C A4 ^ B BN C E EN ^ A5 uN dx 2 R3 2 R3 Z 1 d .uQ ^ B / jN C uN ^ BN jQ dx: C o.1/w-L1 .dt / C o.1/L1.dt / loc dt 2 R3 Now, for mere convenience of notation, we introduce the integrand

I D uQ uN C

1 E EN C B BN 2

0 1 a26 a35 1 1 N E EN ^ B BN uN C @a34 a16 A u: C 2 2 a a 15 24

Thus, further employing the identity (12.29), we find that Z Z Z C d Q I dx C qO C qO MM dvdv d W rx uN dx dt R3 R3 R3 R3 S2 N W 1;1 .dx/ C k@t uk N L1 .dx/ C jNL1 .dx/ kuk C ıH .t/ 1 kuk N L1 .dx/ Z 1 A1 uQ C A4 E C uN ^ B BN C A5 B C E EN ^ uN dx 2 R3 d C o.1/w-L1 .dt / C o.1/L1.dt / loc dt Z 1 C R;11 EN .E C uQ ^ B / jN EN C uN ^ BN jQ dx; 2 R3

352


whence, in view of the estimate (12.28), Z

C Q qO C qO MM dvdv d W rx uN dx R3 R3 R3 R3 S2 N L1 .dx/ C jN L1 .dx/ N W 1;1 .dx/ C k@t uk kuk ıH .t/ C 1 kuk N L1 .dx/ Z 1 A1 uQ C A4 E C uN ^ B BN C A5 B C E EN ^ uN dx 2 R3 d C o.1/w-L1 .dt / C o.1/L1.dt / loc dt Z 1 .E C uQ ^ B / jN C EN C uN ^ BN jQ dx: 2 R3

d dt

Z

Z

I dx C

Using then the approximate Ohm’s law (12.23) and reorganizing the resulting inequality so that all remainder terms appear on its right-hand side, we obtain Z

C Q I dx C qO C qO MM dvdv d W rx uN dx R3 R3 R3 R3 S2 ! Z Z qO C; qO ;C vMM dvdv d jN dx 2 R3 R3 R3 S2 N L1 .dx/ C jNL1 .dx/ N W 1;1 .dx/ C k@t uk kuk ıH .t/ C 1 kuk N L1 .dx/ Z 1 A1 uQ C A4 E C uN ^ B BN C A5 B C E EN ^ uN dx 2 R3 ! Z Z qOC; qO;C vMM dvdv d dx A3 C 2 R3 2 R3 R3 S2 Z 1 d C o.1/w-L1 .dt / C .R;9 C R;10 / jN dx o.1/L1.dt / loc dt 2 R3 ! Z Z C; ;C q O q O ? EN C uN ^ BN P vMM dvdv d dx C 2 R3 2 R3 R3 S2 ! ! Z Z 1 qO C; qO ;C vMM dvdv d C jQ EN C uN ^ BN dx: 2 R3 2 R3 R3 S2

d dt

Z

Z


353

Thus, in view of the convergences (12.32), (12.33), the estimates (12.24), (12.25) and Lemma 12.1 (allowing to control the energy by the entropy), we infer that Z Z Z C d Q qO C qO MM dvdv d W rx uN dx I dx C dt R3 R3 R3 R3 S2 ! Z Z qO C; qO ;C vMM dvdv d jN dx 2 R3 R3 R3 S2 N L1 .dx/ C jNL1 .dx/ N W 1;1 .dx/ C k@t uk kuk ıH .t/ C 1 kuk N L1 .dx/ Z 1 A1 uQ C A4 E C uN ^ B BN C A5 B C E EN ^ uN dx 2 R3 ! Z Z qOC; qO ;C A3 vMM dvdv d dx C 2 R3 2 R3 R3 S2 C o.1/w-L1

loc .dt /

C

d o.1/L1.dt / : dt

(12.36) At last, we may now combine the inequalities (12.35) and (12.36) to deduce, employing the symmetries of collision integrands and (12.12) to rewrite dissipation terms, that 0 2 Z d @ gC C C g jvj gN C E EN C B BN dt R3 K 0 11 1 0 a26 a35 C @ E EN ^ B BN C @a34 a16 AA uN A dx a15 a24 Z C C 1 qO qN C qO qN C qOC; qN C; C qO ;C qN ;C MM dxdvdv d C 2 R3 R3 R3 S2 1 0 uQ 3 Q C B Z Q C B R C;2 ;C C B A B 2 R3 R3 S2 qO qO .t/ıH .t/ vMM dvdv d C dx R3 A @ N E C uN ^ B B B C E EN ^ uN C o.1/w-L1

loc .dt /

C

1 d o.1/L1.dt / ıD .t/: dt 2

354


It may be useful to note here that ıH .t/ o.1/L1.dt / , which follows from Lemma 12.1 and implies that C1 ıH .t/ C2 ıH .t/ C o.1/L1.dt / , for any 0 < C1 C2 . Indeed, up to the introduction of vanishing remainders in L1 .dt/, this conveniently allows us to define the coefficient .t/ with one independent constant only. Next, assembling the preceding inequality with the scaled entropy inequality (12.18) and the energy estimate (12.34), we finally obtain d ıH .t/ C ıD .t/ dt .t/ıH .t/ 0

1 uQ uN 3 Q C B Z Q 52 N 2 C B R C; ;C C AB 2 R3 R3 S2 qO qO vMM dvdv d jN C C dx B R3 A @ N N E E C uN ^ B B N N B B C E E ^ uN 1 d C o.1/w-L1 .dt / C o.1/L1.dt / C ıD .t/; loc dt 2

which, with a straightforward application of Grönwall’s lemma (carefully note that this is valid even though ıH .t/ may be negative), concludes the proof of the proposition. Remark. The proof of Proposition 12.2 is based on the construction of the stability inequality (3.32) from Proposition 3.13 for the two-fluid incompressible Navier– Stokes–Maxwell system with solenoidal Ohm’s law (3.6). This approach has the great advantage of using the approximate macroscopic conservation of momentum established in Proposition 9.6 rather than the one from Proposition 9.5. Indeed, if we were to use the latter approximate conservation law from Proposition 9.5, we would have to modulate the nonlinear term jQ ^ B into jQ jN ^ B BN (much like in the proof of Proposition 3.11; see (3.29)). The term B BN would then have to be absorbed (through Grönwall’s lemma) by a renormalized modulated energy (or entropy), whereas jQ jN would need to be controlled by a renormalized modulated entropy dissipation provided jQ is replaced by the collision integrands ! Z qOC; qO;C vMM dvdv d : 2 R3 R3 S2 However, this last step produces remainders which may not belong to L2 .dtdx/ and, therefore, cannot multiply B . Thus, this procedure would fail. It is therefore not possible (at least, we do not know how to make it work) to establish a similar renormalized relative entropy inequality for renormalized solutions


355

of the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) based on the construction of the stability inequality (3.27) from Proposition 3.11. Using the strategy of Proposition 3.13 removes this difficulty altogether by expressing the Lorentz force jQ ^ B with the Poynting vector E ^ B (and some other terms). However, the drawback of this approach resides in the necessity of the restriction kuk N L1 < 1. Recall, nevertheless, that this restriction is physically relevant, t;x since it merely entails that the modulus of the velocity uN be less than the speed of light (see comments after the proofs of Propositions 3.10 and 3.13).

12.2.4 Convergence and conclusion of the proof We may now pass to the limit in the approximate stability inequality (12.17) and thus derive the crucial modulated energy inequality for the limiting system (4.39). To this end, we simply integrate (12.17) in time against non-negative test functions and then let ! 0, which yields, thanks to the well-preparedness of the initial data (4.41), the weak convergences (12.1), (12.3) and the lower semi-continuities (12.13), (12.14), that Z Rt 1 t ıE .t/ C ıD .s/e s . /d ds 2 0 ıE .0/e

Rt 0

.s/ds

1 u uN 3 C B Z tZ 52 N C B R C;2 ;C C .s/dx B N q dvdv d j q vMM C A B 2 R3 R3 S2 C R3 0 N N A @ E E C uN ^ B B N N B B C E E ^ uN e

Rt s

. /d

0

ds:

Further observe that the characterization (6.11) of the limiting collision integrands q ˙; from Proposition 6.3 implies, using the definition (2.75) of the constant , Z C; q ;C vMM dvdv d q 2 R3 R3 S2 2 Z jvj j L .v/ C wL D vM dv 2 R3 2 Z v L .v/ M dvj D j: D 6 R3

356


Therefore, using (12.2), we deduce that Z Rt 1 t ıE .t/ C ıD .s/e s . /d ds 2 0 0 ıE .0/e

Rt 0

.s/ds

1 u uN 5 C B Z tZ N Rt 2 C B C .s/dx e s . /d ds; B N j j C AB C R3 0 @E EN C uN ^ B BN A B BN C E EN ^ uN

which is precisely the stability inequality we were after. As for the temporal continuity of u; 52 ; E; B , it is readily seen from the approximate macroscopic conservation laws from Proposition 9.5 and Maxwell’s equations (12.27) that @t P uQ , @t 32 Q Q , @t E and @t B are uniformly bounded, in L1loc in time and in some negative index Sobolev space in x. It is therefore pos sible to show (see [59, Appendix C]) that P uQ ; 32 Q Q ; E ; B converges to 5 u; 2 ; E; B 2 C.Œ0; 1/I w-L2 .R3 // weakly in L2 .dx/ uniformly locally in time. At last, the proof of Theorem 4.6 is complete.

12.3 Proof of Theorem 4.7 on strong interactions This proof closely follows the method of proof of Theorem 4.6 presented in the preceding section. However, the asymptotic limit treated in Theorem 4.7 is more singular than the one from Theorem 4.6. Some steps in the coming proof will therefore require some greater care than their counterparts from the previous section. As before, we begin our proof by appropriately gathering previous results together.

12.3.1 Weak convergence of fluctuations, collision integrands and electromagnetic fields Thus, we are considering here a family of renormalized solutions .f˙ ; E ; B / to the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35), in the regime of strong interspecies interactions, i.e., ı D 1, satisfying the scaled entropy inequality (4.36). By Lemmas 5.1 and 5.2, the corresponding families of fluctuations g˙ and renormalized fluctuations gO ˙ are weakly compact in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and L2loc .dtI L2 .Mdxdv//, respectively, while by Lemma 5.3, the corresponding collision integrands qO ˙ and qO˙; are weakly compact in L2 .MM dtdxdvdv d /. Thus, using Lemma 5.1 again and the decomposition (5.5), we know that there exist g ˙ 2 L1 .dtI L2 .Mdxdv//, .E; B/ 2 L1 .dtI L2 .dx// and q ˙ ; q ˙; 2

357

12.3 Proof of Theorem 4.7 on strong interactions

L2 .MM dtdxdvdv d /, such that, up to extraction of subsequences, in L1loc dtdxI L1 1 C jvj2 Mdv ; g˙ * g ˙ in L1 dtI L2 .Mdxdv/ ; gO ˙ * g ˙ .E ; B / * .E; B/ in L1 dtI L2 .dx/ ; qO ˙ * q ˙ in L2 MM dtdxdvdv d ; in L2 MM dtdxdvdv d ; qO ˙; * q ˙;

(12.37)

as ! 0. Therefore, one also has the weak convergence of the densities ˙ , bulk ˙ ˙ velocities u˙ and temperatures corresponding to g : ˙ * ˙ ;

u˙ * u˙

and ˙ * ˙


where ˙ ; u˙ ; ˙ 2 L1 .dtI L2 .dx// are, respectively, the densities, bulk velocities and temperatures corresponding to g ˙ . In fact, Lemma 5.10 implies that 2 3 ˙ ˙ ˙ ˙ ˙ jvj g D …g D C u v C : 2 2 Next, we further introduce the scaled fluctuations h D

1 C g g n ;

where n D C is the charge density, and the electrodynamic variables j D

1 C u u ;

w D

1 C ;

which are precisely the bulk velocity and temperature associated with the scaled fluctuations h . In view of Lemma 5.13, the electric current j and the internal electric energy w are uniformly bounded in L1loc .dtdx/, which necessarily implies, letting ! 0, that uC D u and C D . Carefully note, though, that the limiting densities C and may be distinct here. Therefore, we appropriately rename the limiting macroscopic variables D whence

C C ; 2

n D C ;

u D uC D u ;

jvj2 3 g D CuvC ; 2 2 2 3 gC C g jvj DCuvC ; 2 2 2 g C g D n: ˙

˙

D C D ;

358


Now, according to Lemma 5.14, it holds that h C 1 C gO gO

L2 .Mdv/

is weakly compact in L1loc .dtdxI L1 ..1 C jvj/Mdv// and that j 1C

kgO C

gO kL2 .Mdv/

is weakly compact in L2loc .dtdx/. Moreover, Lemma 7.11 indicates that, up to extraction of subsequences, there is an h 2 L1loc .dtdxI L1 ..1 C jvj2 /Mdv// such that h C 1 C gO gO

* L2 .Mdv/

in L1loc dtdxI L1 ..1 C jvj/Mdv/ ;

h 1 C jnj

as ! 0. Note, however, that h is not characterized by an infinitesimal Maxwellian form. Here, we only have that 2 3 jvj ; …h D j v C w 2 2 where the electric current j and the internal electric energy w are defined by 2 Z Z jvj hvM dv; wD h 1 M dv: j D 3 R3 R3 In particular, j C 1 C gO gO

*

L2 .Mdv/

j 1 C jnj

in L2loc .dtdx/:

(12.38)

Finally, since n 2 L1 .dtI L2 .dx//, setting r D

1 C jnj C 2 L1 dtI L2loc .dx/ ; 1 C gO gO L2 .Mdv/

which, according to Lemma 7.11 and up to extraction of subsequences, converges almost everywhere towards the constant function 1, we see that r j * j

in L1loc .dtdx/:

Similarly, since h C 1 C gO gO

L2loc .dtdxI L1 ..1

is bounded in one can show that

L2 .Mdv/

C jvj/Mdv// uniformly, by virtue of Lemma 5.14,

r h * h in L1loc dtdxI L1 ..1 C jvj/Mdv/ :

359


12.3.2 Constraint equations, Maxwell’s system and the energy inequality In view of Proposition 8.3, we already know that the limiting thermodynamic fields , u and satisfy the incompressibility and Boussinesq relations div u D 0;

C D 0:

(12.39)

Moreover, the discussion in Section 6.4 shows that the limiting electromagnetic field satisfies the Faraday equation and Gauss’ laws: 8 ˆ < @t B C rot E D 0; div E D n; ˆ : div B D 0: Recall, however, that we do not know from Section 6.4 whether Ampère’s equation is necessarily satisfied in the sense of distributions in the limit. Finally, Proposition 8.5 further establishes that the electrodynamic variables j and w satisfy Ohm’s law and the internal electric energy equilibrium relation 1 j nu D rx n C E C u ^ B ; w D n; 2 where the electric conductivity > 0 is defined by (2.70). As for the energy bound, Proposition 8.6 states that, for almost every t 0, 1 1 knk2L2 C 2 kuk2L2 C 5 k k2L2 C kEk2L2 C kBk2L2 .t/ x x x x x 2 2 Z t 1 2 2 2 C 2 krx ukL2 C 5 krx kL2 C kj nukL2 .s/ ds C in ; x x x 0 where the viscosity > 0, thermal conductivity > 0 and electric conductivity > 0 are respectively defined by (2.61) and (2.70). In particular, it holds that .n; u; ; E; B/ 2 L1 Œ0; 1/; dtI L2 R3 ; dx ; .u; / 2 L2 Œ0; 1/; dtI HP 1 R3 ; dx ; j nu 2 L2 Œ0; 1/ R3 ; dtdx : This energy bound can be improved to the actual energy inequality 1 1 2 2 2 2 2 knkL2 C 2 kukL2 C 5 k kL2 C kEkL2 C kBkL2 .t/ x x x x x 2 2 Z t 1 C 2 krx uk2L2 C 5 krx k2L2 C kj nuk2L2 .s/ ds x x x 0 in 2 1 1 in 2 in 2 in 2 in 2 n L2 C 2 u L2 C 5 L2 C E L2 C B L2 ; x x x x x 2 2 using the well-preparedness of the initial data (4.44).

360


12.3.3 The renormalized modulated entropy inequality We move on now to the rigorous derivation of a stability inequality encoding the asymptotic macroscopic evolution equations for u and and the Ampère equation in the spirit of the weak-strong stability inequalities used in Section 3.2.3 to define dissipative solutions for some Navier–Stokes–Maxwell systems. Recall that, as explained therein, such systems are in general not known to display weak stability, so that their weak solutions in the energy space are not known to exist. The strategy used here closely follows the method employed in the case of weak interactions detailed in Section 12.2.3. 2 Thus, as in Section 9.2, we define the renormalized fluctuations g˙ ˙ jvj , K 1 with K D Kj log j, for some large K > 0, and 2 Cc .Œ0; 1// a smooth compactly supported function such that 1Œ0;1 1Œ0;2 , and where ˙ D .G˙ / for some renormalization 2 C 1 .Œ0; 1/I R/ satisfying (9.1). We also consider here an auxiliary renormalization 2 C 1 .Œ0; 1/I R/ satisfying (9.36), which will be used later on when applying estimates from Section 9.2, and denote ˙ D .G˙ /. jvj2 ˙ Since, up to further extraction of subsequences, K converges almost everywhere towards 1, g˙ is weakly compact in L1loc .dtdxI L1 ..1 C jvj2 /Mdv// and g˙ ˙ is uniformly bounded in L1 .dtI L2 .Mdxdv//, we deduce, by the Product Limit Theorem, that 2 jvj * g ˙ in L1 dtI L2 .Mdxdv/ : g˙ ˙ K Similarly, g˙ ˙

jvj2 K

* g ˙

in L1 dtI L2 .Mdxdv/ :

Therefore, one has the weak convergence of the densities Q˙ , bulk velocities uQ ˙ and 2 temperatures Q˙ corresponding to g˙ ˙ jvj : K

Q˙ * ˙ ;

uQ ˙ * u and Q˙ *

In particular, the hydrodynamic variables Q D QC CQ 2

in L1 dtI L2 .dx/ as ! 0: C

Q CQ , 2

uQ D

C

u Q Cu Q 2

and Q D

also obviously verify

Q * ;

uQ * u and Q *

in L1 dtI L2 .dx/ as ! 0;

while the charge density nQ D QC Q satisfies nQ * n in L1 dtI L2 .dx/ as ! 0:

(12.40)

(12.41)


361

It follows that, since u is solenoidal, P ? uQ * 0 in L1 dtI L2 .dx/ as ! 0; and, in view of the limiting Boussinesq relation, Q C Q * 0 in L1 dtI L2 .dx/ as ! 0:

(12.42)

(12.43)

Here, in constrast with the convergence properties of the electric current established in Section 12.2.3 for weak interactions, we cannot show the convergence of the u Q electric current jQ D 1 uQ C towards j unless we renormalize it as in (12.38). Instead, we establish below in (12.64) a useful consistency relation for jQ by suitably controlling remainders in the spirit of Section 9.2. 2 Now, just as2 in the case of weak interspecies interactions, the L .Mdxdv/ norm of g˙ ˙ jvj is not a Lyapunov functional but it is nevertheless controlled by the K relative entropy 2 2 C 1 ˙ ˙ g jvj 2 H f˙ ; (12.44) 2 K L2 .Mdxdv/

for some C > 1, and therefore by the initial data (4.45). One may therefore try, in a preliminary attempt to show an asymptotic inequality, to modulate the stability

jvj2 ˙ ˙ approximate energy associated with g K , i.e., its L2 .Mdxdv/ norm, by introducing a test functions gN ˙ in infinitesimal Maxwellian form: 2 3 jvj ˙ ˙ N gN D N C uN v C ; 2 2 where C N x/ 2 C 1 Œ0; 1/ R3 with div uN D 0; N C N C N D 0; N x/; .t; N˙ .t; x/; u.t; c 2

and then establishing a stability inequality for the modulated energies 2 2 1 g ˙ ˙ jvj gN ˙ : 2 2 K L .Mdxdv/ Notice that the elementary identity 3 3 5 a2 C b 2 D .a C b/2 C 2 5 2

3b 2a 5

2 ;

(12.45)

362


for any a; b 2 R, yields that 2 2 X ˙ ˙ g jvj gN ˙ 2 K

˙

L .Mdxdv/

2 2 X jvj ˙ ˙ ˙ gN … g K L2 .Mdxdv/ ˙ X ˙ 2 3 Q ˙ N 2 ˙ ˙ 2 Q N L2 .dx/ C uQ uN L2 .dx/ C 2 D L .dx/ 2 ˙

1 N 2L2 .dx/ C knQ nk N 2L2 .dx/ D 2 kQ k 2 X 2 3 Q ˙ N 2 ˙ uQ uN L2 .dx/ C 2 C L .dx/ 2 ˙

1 N 2L2 .dx/ D knQ nk 2 0 2 3Q ˙ 2Q 2 X 3 5 2 uQ ˙ uN 2 N @ Q C Q˙ C C C L .dx/ 2 L2 .dx/ 5 2 5 ˙

1 A;

L .dx/

C

N where we have denoted N D N C and nN D NC N . 2 As before, it turns out that this approach is not quite suitable for our purpose because, even though, for any 0 t1 < t2 (see the proof of Lemma 5.1), Z t2 Z t2 1 1 ˙ g ˙ 2 2 dt lim inf H f dt; (12.46) L .Mdxdv/ 2 !0 t1 2 t1

it is not possible to set C D 1 in (12.44). Indeed, the first term in the polynomial expansion of the function h.z/ D .1 C z/ log.1 C z/ z defining the entropy is 12 z 2 , but the second term is 16 z 3 and may be negative. Some entropy (or energy) is therefore lost by considering the modulated energies (12.45). These considerations lead us to introduce a more precise modulated functional in replacement of (12.45), capturing more information on the fluctuations. To be precise, instead of (12.45), we consider now the renormalized modulated entropies 2 Z 1 2 1 ˙ jvj ˙ ˙ gN ˙ M dxdv C gN ˙ L2 .Mdxdv/ : (12.47) H f g 2 K 2 R3 R3 Note that the above functional may be negative for fixed > 0. However, in view of (12.46), it recovers asymptotically a non-negative quantity, which is precisely the

363


asymptotic modulated energy: Z t2 2 3 1 2 ˙ N˙ 2 2 N dt C uk N C ku L2 .dx/ L .dx/ L2 .dx/ 2 t1 2 Z t2 1 g ˙ gN ˙ 2 2 D dt L .Mdxdv/ t1 2 Z t2 1 ˙ lim inf H f !0 2 t1 2 Z 1 2 jvj gN ˙ M dxdv C gN ˙ L2 .Mdxdv/ dt; g˙ ˙ K 2 R3 R3

(12.48)

for all 0 t1 < t2 . The first term in (12.47) is precisely the entropy of f˙ and will be controlled by the scaled entropy inequality (4.36), whereas the last term in (12.47) only involves smooth quantities and will therefore be controlled directly. As for the middle term in the modulated entropy (12.47), its time derivative 2 will involve the approximate ˙ ˙ macroscopic conservation laws for g jvj . K Now, in order to establish the renormalized modulated entropy inequality leading to the convergence stated in Theorem 4.7, we introduce further test functions N x/; B.t; N x/; jN .t; x/ 2 Cc1 Œ0; 1/ R3 with div EN D n; N div BN D 0; E.t; and we define the renormalized modulated entropy ıH .t/ D

1 C 1 H f C 2 H f 2 2 2 Z jvj jvj C C C g gN C g gN M dxdv K K R3 R3 1 1 2 C gN C L2 .Mdxdv/ C kgN k2L2 .Mdxdv/ 2 2 2 2 1 1 C E EN L2 .dx/ C B BN L2 .dx/ 2Z 2 1 1 C Tr m C Tr a dx 2 R3 2 0 11 0 Z a26 a35 @ E EN ^ B BN C @a34 a16 AA uN dx; R3 a15 a24

where the matrix measures m and a are the defects introduced in Section 4.1.4 and controlled by the scaled entropy inequality (4.36).

364


We also define the renormalized modulated energy 2 2 1 jvj C C C gN ıE .t/ D g 2 K L2 .Mdxdv/ 2 2 1 jvj gN C g 2 K L2 .Mdxdv/ 2 1 1 2 C E EN L2 .dx/ C B BN L2 .dx/ 2Z 2 1 1 C Tr m C Tr a dx 2 R3 2 0 0 11 Z a26 a35 @ E EN ^ B BN C @a34 a16 AA uN dx; R3 a15 a24 which is asymptotically equivalent to ıH .t/, at least formally. Note that ıH .t/ controls more accurately the large values of the fluctuations g˙ than ıE .t/. Lemma 12.3 below shows how the modulated entropy ıH .t/ controls the modulated energy ıE .t/. Finally, we introduce the renormalized modulated entropy dissipation 2 1 1 ıD .t/ D qOC qN C L2 MM dxdvdv d C kqO qN k2 2 L MM dxdvdv d 4 4 2 1 C qO C; qN C; L2 MM dxdvdv d 4 1 2 C qO ;C qN ;C L2 MM dxdvdv d ; 4 where 1 1 qN ˙ D rx uN W Q C Q Q 0 Q 0 C rx N Q C Q Q 0 Q 0 2 2 1 N 0 Q ˆ Q 0 ; Q Cˆ Qˆ j nN uN ˆ (12.49) 1 1 ˙; 0 0 0 0 N Q Q Q Q Q Q Q Q C D rx uN W C C rx qN 2 2 1 N Q0 Cˆ Q 0 ; Q ˆ Qˆ j nN uN ˆ so that Z C 1 qN C qN C qN C; C qN ;C M dv d D rx uN W L Q C rx N L Q 2 R3 S2 D rx uN W C rx N ; Z C 2 Q ; qN qN M dv d D jN nN uN L ˆ R3 S2


365

Z

C; 2 Q ; qN qN ;C M dv d D jN nN uN L ˆ R3 S2 qN C C qN qN C; qN ;C D 0;

Q defined by (2.68). with , , Q and Q defined by (2.14) and (2.15) and ˆ Then, assuming from now on that kuk N L1 .dt dx/ < 1 and using the lower weak sequential semi-continuity of the entropies (12.48) and of the electromagnetic energy (3.25) together with Lemma 4.4, we find that, for all 0 t1 < t2 , Z t2 Z t2 Z t2 0 ıE .t/ dt lim inf min ıH .t/ dt; ıE .t/ dt ; (12.50) !0

t1

t1

t1

where ıE .t/ 2 1 1 D g C gN C L2 .Mdxdv/ C kg gN k2L2 .Mdxdv/ 2 2 Z 2 2 1 1 C E EN L2 .dx/ C B BN L2 .dx/ E EN ^ B BN uN dx 2 2 R3 2 1 3 N 2L2 .dx/ C ku uk N 2L2 .dx/ C N L2 .dx/ D k k N 2L2 .dx/ C kn nk 4 2 Z 2 2 1 1 C E EN L2 .dx/ C B BN L2 .dx/ E EN ^ B BN uN dx 2 2 R3 2 5 1 N 2L2 .dx/ C ku uk N 2L2 .dx/ C N L2 .dx/ D kn nk 4 2 Z 2 2 1 1 N N C E EN ^ B BN uN dx; E E L2 .dx/ C B B L2 .dx/ 2 2 R3 while, repeating mutatis mutandis the computations leading to (8.47) and (8.49) in the proof of Proposition 8.6, we obtain, for all 0 t1 < t2 , Z t2 Z t2 ıD .t/ dt lim inf ıD .t/ dt; (12.51) t1

where

!0

t1

2 N 2L2 C 5 rx N L2 ıD .t/ D 2 krx .u u/k x x 2 1 C .j nu/ jN nN uN L2 x 1 1 2 q C qN C L2 MM dxdvdv d C kq qN k2 2 L MM dxdvdv d 4 4 2 1 C q C; qN C; L2 MM dxdvdv d 4 1 2 C q ;C qN ;C L2 MM dxdvdv d : 4

366


The following lemma shows how the modulated entropy ıH controls the modulated energy ıE up to a small remainder. It is obtained by repeating the proof of Lemma 12.1 and, thus, we skip the details of its proof. Lemma 12.3. It holds that ıE .t/ C ıH .t/ C o.1/L1.dt / ; for some fixed constant C > 1. The following result establishes the renormalized modulated entropy inequality at the order , which will eventually allow us to deduce the crucial weak-strong stability of the limiting thermodynamic fields, thus defining dissipative solutions. Proposition 12.4. One has the stability inequality Z Rt 1 t ıD .s/e s . /d ds ıH .t/ C 2 0 Rt

ıH .0/e 0 .s/ds Z t Z C A R3 0 0

1 uQ uN 3 Q C B Q 52 N C B 2 C B R .s/ dx B 3 3 2 qO C qO C qO C; qO ;C ˆMM Q N dvdv d j nN uN C C B R R S 1 A @ E EN C uN ^ B BN 2 rx .nQ n/ N B BN C E EN ^ uN

e

Rt s

. /d

ds C o.1/L1 ; loc .dt /

(12.52)

where the acceleration operator is defined by 1 A1 B A2 C C N n; N BN D B A u; N ; N jN ; E; B A3 C @A A 4 A5 1 0 2 .@t uN C P .uN rx u/ N x u/ N C P nN EN C jN ^ BN B N N N C B C 2 @t C uN1 rx x 1 N B C; N N DB j nNuN 2 rx nN C E C u N ^ B C @ A N N N @tE rot B Cj N N @t B C rot E 0


367

and the growth rate is given by

.t/ D C

N jN nN uN .t/ 1 N N ku.t/k W 1;1 .dx/ C k@t u.t/k L1 .dx/ C .t/ W 1;1 .dx/ C L .dx/ 1 ku.t/k N L1 .dx/ ! 2 1 N 1;1 N N ; C .t/ C 2 rx nN E uN ^ B .t/ 1 W .dx/ L .dx/

with a constant C > 0 independent of test functions and . Proof. The main ingredients of the proof of this stability inequality are: The scaled entropy inequality (4.36): Z 1 C 1 1 H f H f Tr m dx C C 2 2 2 2 R3 Z 1 C jE j2 C jB j2 C Tr a dx 2 R3 Z Z 2 2 1 t 2 qO C C qO C qO C; C 4 0 R3 R3 R3 S2 2 C qO;C MM dxdvdv d ds Z in 2 1 Cin 1 in 1 2 H f C 2 H f C jE j C jBin j2 dx; 2 R3

(12.53)

which is naturally satisfied by renormalized solutions of the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) (provided they exist) and where we have used the inequality (5.7) from Lemma 5.3 in order to conveniently simplify the dissipation terms. The approximate conservation of energy obtained in Proposition 9.5: Z qO C C qO Q 3Q 5 Q Q C rx uQ MM dvdv d @t 2 2 2 R3 R3 S2 D R;3 : (12.54) The remainder R;3 satisfies 1 C ıE .t/ C C ıE .t/ıD .t/ 2 C o.1/L1 .dt / ; loc 1 kuk N L1 .dx/ (12.55) for some C > 0, where we have used (12.26) to bound kR;3 kW 1;1 .dx/ loc

E EN 2 2

L .dx/

C ıE .t/ ; 1 kuk N L1 .dx/

368


since ıE is not a sum of only non-negative terms. Note that we do not employ the approximate conservation of momentum from Proposition 9.5, which is crucial (see comments following the present proof, below). The approximate conservation of momentum law from Proposition 9.6 0 11 0 a26 a35 1 1 @t @uQ C E ^ B C @a34 a16 AA 2 2 a a 15

24

1 juQ j2 Id C 2 m 3 2

Crx uQ ˝ uQ

! qO C C qO Q MM dvdv d 2 R3 R3 S2 1 jE j2 C jB j2 C Tr a rx .E ˝ E C e C B ˝ B C b / C rx 2 4

1 D rx Q C Q C @t .R;8 / C R;7 ; (12.56) where the remainders R;7 and R;8 satisfy Z

R;8 D o.1/L1dt IL1

;

loc .dx/

kR;7 kW 1;1 .dx/ C1 ıH .t/ C C2 ıE .t/ C o.1/L1

; loc .dt /

loc

for some C1 ; C2 > 0. The approximate Ohm’s law Z C 1 Q qO qO C qO C; qO ;C ˆMM dvdv d R3 R3 S2 1 D rx nQ .E C uQ ^ B / C R;9 C rx R;10 C R;11 ; 2

(12.57)

(12.58)

where > 0 is defined by (2.70) and the remainders R;9 and R;10 vanish weakly R;9 D o.1/w-L1

loc .dt dx/

and R;10 D o.1/w-L1

loc .dt dx/

;

(12.59)

whereas R;11 satisfies kR;11 kL1 .dx/

C ıE .t/ C o.1/w-L1 .dt / : loc 1 kuk N L1 .dx/

(12.60)

This approximate law is obtained directly from the limiting laws derived in Proposition 8.3. Indeed, it is easily deduced from (8.32) that (12.58) holds

369


with the remainders Z C 1 Q R;9 D qO qO C qOC; qO;C ˆMM dvdv d R3 R3 S2 Z C 1 Q q q C q C; q ;C ˆMM dvdv d R3 R3 S2 C .E E/ C .uQ u/ ^ B C u ^ .B B/ ; 1 R;10 D .nQ n/ ; 2 O .ıE .t/ C ıE .t//L1 .dx/ R;11 D .uQ u/ ^ .B B/ D ; 1 kuk N L1 .dx/ where we have used (12.26). The above estimate on R;11 is then readily improved to (12.60) upon noticing from (12.50) that ıE .t/ ıE0 .t/; where ıE0 .t/ is the limit, up to extraction of subsequences, of ıE .t/ in w -L1 .dt/, and then writing ıE .t/ C ıE .t/ 2ıE .t/ C ıE0 .t/ ıE .t/ D 2ıE .t/ C o.1/w-L1 .dt / : As in the case of the approximate solenoidal Ohm’s law (12.23) for weak interspecies interactions, it would be possible to derive the above approximate Ohm’s law employing the methods of proof of Proposition 9.5. Nevertheless, the method presented here is more robust. Maxwell’s equations: 8 @t E rot B ˆ ˆ ˆ < @t B C rot E ˆ div E ˆ ˆ : div B

D j D jQ C R;12 ; D 0; D n D nQ R;13 ; D 0;

(12.61)

where the remainders R;12 D jQ j and R;13 D nQ n satisfy kR;12 kL1

loc .dx/

and

C ıH .t/ C o.1/L1

loc .dt /

R;13 D o.1/L1dt IL1

:

loc .dx/

;

(12.62)

(12.63)

The convergence (12.63) straightforwardly follows from (9.63). As for the control (12.62), it is obtained through the following estimate. First, since G˙ 2

370


p and gO ˙ 2. 2 1/ on the support of 1 ˙ , we easily deduce, using Lemma 9.9, that 1 ˙ g 1 ˙ 1 2 L1 loc dxIL .1Cjvj/ Mdv 1 ˙ ˙2 ˙ D gO C gO 1 4 1 1 2 2 ˙ o n C 1 G ˙ 2 gO 2

Lloc dxIL

Lloc dxIL2 .1Cjvj/2 Mdv

.1Cjvj/ Mdv

2 2 ! 1 ˙ 1 jvj M dxdv h g C1 g˙ ˙ 2 3 3 2 K R R 2 2 ˙ ˙ jvj ˙ C C2 g N C o.1/L1 .dt / g 2 loc K L .Mdxdv/ Z

C1 ıH .t/ C C2 ıE .t/ C o.1/L1

loc .dt /

:

Furthermore, using the Gaussian decay (8.8) and that g˙ ˙ is comparable to gO , we also obtain 2 1 ˙ ˙ jvj g 1 1 K L .1Cjvj/2 Mdv 2 1 ˙ jvj C gO 1 1 K L .1Cjvj/2 Mdv 2 1 jvj 1 C gO ˙ L2 .Mdv/ 2 K L .1Cjvj/4 Mdv 5

5 K : D O K 4 jlog j 4 4 1 1 2 L

dt IL .dx/

Thus, further using Lemma 12.3, we infer, provided K > 4, that 2 1 g ˙ g ˙ ˙ jvj C ıH .t/Co.1/L1loc.dt / ; K L1 dxIL1 .1Cjvj/2 Mdv loc

whence

jQ j 1 L

loc .dx/

C ıH .t/ C o.1/L1

loc .dt /

;

(12.64)

which establishes (12.62). Notice that we cannot rigorously write the identities (1.10) and (1.12) for the above system, because the source terms j and n do not belong to L2loc .dtdx/


a priori. Nevertheless, one has the following modulated identities: @t E EN C B BN C rx E ^ BN C EN ^ B D jQ R;12 EN jN C A4 E A5 B ;

371

(12.65)

and @t

ˇ2 ˇ ˇ2 1 ˇ E EN ^ B BN C rx ˇE EN ˇ C ˇB BN ˇ 2 N rx E E ˝ E EN C B BN ˝ B BN

1 D @t .E ^ B / C rx jE j2 C jB j2 rx .E ˝ E C B ˝ B / 2 C jN C A4 ^ B BN C E EN ^ A5 C jQ R;12 ^ BN C .nQ R;13 / EN C nN E EN : (12.66)

Finally, taking the divergence of the approximate Ampère equation from (12.61), we obtain the approximate conservation of charge (or approximate continuity equation) @t nQ C rx jQ D @t R;13 C rx R;12 :

(12.67)

Note that we could just as well use the approximate conservation of charge from Proposition 9.5. The asymptotic characterization (8.31) of the limiting collision integrands from Proposition 8.3 combined with (8.40) from Proposition 8.4, which implies that Z qO C C qO Q MM dvdv d rx uQ C rxt uQ ! 0; 2 R3 R3 S2 Z 5 qO C C qO Q MM dvdv d rx Q ! 0; 3 3 2 2 2 R R S qO C C qO qO C; qO ;C ! 0; (12.68) in the sense of distributions, where ; > 0 are defined by (2.61). The asymptotic characterizations (8.38) and (8.39) of the limiting collision integrands from Proposition 8.4, whose proofs imply that Z C Q Q Q uQ D R;14 ; qO qO C qO C; qO ;C ˆMM dvdv d C j n R3 R3 S2

(12.69)

where the remainder R;14 satisfies kR;14 kL1

loc .dx/

C ıH .t/ C o.1/L1

loc .dt /

:

(12.70)

372


Indeed, we first obtain from (8.41), using Lemmas 9.10 and 12.3, that h hO 1 nO gO C OC C gO O 4 1 L1 1Cjvj2 Mdv (12.71) loc dxIL C ıH .t/ C o.1/L1 .dt / ; loc

where hO D 1 gO C gO nO , nO D OC O and O˙ are the densities associated with the fluctuations gO ˙ . Next, combining (8.44) with (8.45), straightforward computations yield that 1 .L C L/ h nO gO C C gO 2 Z C C qO qO C qO C; qO ;C M dv d R3 S2

D

1 C Q gO gO nO ; gO C C gO 2 1 C C O C .L C L/ h h nO gO O C gO O 4 C 1 nO L gO C gO … gO C C gO ; 2

which implies, using (2.68) and the self-adjointness of L CL and then employing Lemmas 9.7, 9.8 (on consistency estimates) and 12.3 (allowing to control the energy by the entropy) with the estimates (5.14) and (12.71), that Z C C; ;C Q j nO uO C q O C q O q O dvdv d q O ˆMM R3 R3 S2

C ıH .t/ C o.1/L1

loc .dt /

L1 loc .dx/

;

R where uO D 12 R3 gO C C gO vM dv. Finally, utilizing the control (12.64) with yet another application of Lemma 9.7 allows us to deduce the validity of (12.69) from the preceding estimate. Now, by the definition of the acceleration operator A, straightforward energy computations, similar in the proof of Proposition 3.3, applied to the to those performed N BN , show that the following identities holds: test functions u; N N ; n; N jN; E; Z d A1 uN dx D kuk N 2L2 N 2L2 2 krx uk x x 3 dt R Z C nN EN C jN ^ BN uN dx; R3 Z 2 d A2 N dx D kN k2L2 2 rx N L2 ; x x dt R3

373


Z 1 N 1 2 N N A3 j nN uN dx D kj nN uk N L2 C E rx nN jN dx x 2 R3 R3 Z nN EN C jN ^ BN uN dx; R3 Z Z 1 d 2 A4 rx nN dx D jN rx nN dx; knk N L2 x 2 dt R3 R3 Z

Z 1 d N 2 2 N N N kEkL2 C kBkL2 A4 E C A5 B dx D EN jN dx: x x 2 dt R3 R3 Z

All in all, combining the preceding expressions yields the energy identity: 0 1 uN 5 N B C Z B C 2 d N B C N E .t/ C DN .t/ D A B j nN uN C dx; B C 3 dt R @EN 12 rx nN A BN

(12.72)

where the energy EN and energy dissipation DN are defined by 1 2 1 1 2 1 2 EN .t/ D gN C L2 .Mdxdv/ C kgN k2L2 .Mdxdv/ C EN L2 .dx/ C BN L2 .dx/ 2 2 2 2 1 3 2 N 2L2 .dx/ C kuk D kk N 2L2 .dx/ C knk N 2L2 .dx/ C N L2 .dx/ 4 2 2 2 1 1 C EN L2 .dx/ C BN L2 .dx/ ; 2 2 5 2 1 2 1 2 1 2 N L2 .dx/ C kuk N 2L2 .dx/ C N L2 .dx/ C EN L2 .dx/ C BN L2 .dx/ ; D knk 4 2 2 2 and 2 2 1 DN .t/ D 2 krx uk N 2L2 C 5 rx N L2 C jN nN uN L2 x x x 1 2 qN C C qN C qN C; C qN ;C 2 D L MM dxdvdv d 16 2 2 1 1 C qN C qN L2 MM dxdvdv d C qN C; qN ;C L2 MM dxdvdv d 8 8 1 1 2 D qN C L2 MM dxdvdv d C kqN k2 2 L MM dxdvdv d 4 4 2 2 1 1 C qN C; L2 MM dxdvdv d C qN ;C L2 MM dxdvdv d : 4 4 Next, notice that a slight variant of the estimate (12.35) derived in the proof of Proposition 12.2 on weak interactions is also valid here in the case of strong interactions. Indeed, reproducing the very same duality computations preceding (12.35)

374


onto the approximate conservation of energy (12.54) and, then, using the convergences (12.42), (12.43), (12.68), the estimate (12.55) and Lemma 12.3 (allowing to control the energy by the entropy), yields that Z d 3Q Q N dx dt R3 2 Z Z C 1 C; ;C Q C qO C qO C qO C qO MM dvdv d rx N dx 2 R3 R3 R3 S2 ! 1 E .t/ ı C N W 1;1 .dx/ C .ıE .t/ıD .t// 2 1 kuk N L1 .dx/ Z 1 3Q Q dx C o.1/w-L1 .dt / A2 loc 2 R3 2 ! N 1;1 2 1 W .dx/ C C N W 1;1 .dx/ ıH .t/ ıD .t/ 1 kuk N L1 .dx/ 4 Z 1 3Q Q dx C o.1/w-L1 .dt / : A2 loc 2 R3 2 (12.73) Likewise, following the proof of Proposition 12.2, using the solenoidal property div uN D 0, analogous duality computations applied to the approximate conservations of momentum (12.56) and charge (12.67) yield that d dt

Z

1 1 nQ nN C uQ uN C .E ^ B / uN 4 2 1 0 ! 1 1 @a26 a35 A a34 a16 uN R;8 uN R;13 nN dx C 2 a a 4 15 24 0 0 1 1 Z a a 26 35 @ 1 .E ^ B / @t uN C 1 @a34 a16 A @t uN R;8 @t uN 1 R;13 @t nN A dx 3 2 2 a a 4 R 15 24 Z 1 1 C nQ rx jN jQ rx nN C .P uQ / ˝ uN uQ ˝ uQ 2 m W rx uN dx 4 2 R3 Z Z C qO C qO Q C MM dvdv d W rx uN x uN uQ dx 2 R3 R3 R3 S2 Z 1 .E ˝ E C e C B ˝ B C b / W rx uN dx C 2 3 Z R 1 R;7 uN R;12 rx nN dx D 4 R3 Z 1 1 1 N N N C P nN E C j ^ B uQ A1 uQ nQ rx A4 dx; 2 2 4 R3 R3


375

whence, reorganizing some terms so that remainders are moved to the right-hand side, 0 0 1 1 Z d 1 1 a26 a35 1 @ nQ nN C uQ uN C .E ^ B / uN C @a34 a16 A uN A dx dt R3 4 2 2 a a 15 24 Z Z C Q W rx uN dx C qO C qO MM dvdv d R3 R3 R3 S2 Z 1 1 1 R;7 uN R;12 rx nN A1 uQ nQ rx A4 dx D 4 2 4 R3 Z 1 1 N C P nN E C jN ^ BN uQ C jQ rx nN nQ rx jN dx 2 4 R3 Z ? ? 1 uN ˝ P uQ C P uQ ˝ uN C .uQ u/ C N ˝ .uQ u/ N C 2 m W rx uN dx 2 R3 Z Z C qO C qO Q uQ x uN C C MM dvdv d W rx uN dx 2 R3 R3 R3 S2 0 1 0 1 Z 1 a26 a35 1 1 @ .E ^ B / @t uN C @a34 a16 A @t uN R;8 @t uN R;13 @t nN A dx C 2 a a 4 R3 2 15 24 Z 1 .E ˝ E C e C B ˝ B C b / W rx uN dx 2 R3 Z d 1 C R;8 uN C R;13 nN dx: dt R3 4 Then, using the convergences (12.42), (12.68), the estimates (4.20), (4.25), (12.57), (12.63) and Lemmas 4.4 and 12.3 (allowing to control the energy by the entropy), we arrive at 0 1 1 0 Z a a 26 35 d @ 1 nQ nN C uQ uN C 1 .E ^ B / uN C 1 @a34 a16 A uN A dx dt R3 4 2 2 a a 15 24 Z Z C Q N dx qO C qO MM C dvdv d W rx u 3 3 3 2 R R R S ! Z N L1 .dx/ k@t uk 1 ıH .t/ C kuk N W 1;1 .dx/ C R;12 rx nN dx 1 kuk N L1 .dx/ 4 R3 Z 1 d 1 A1 uQ C nQ rx A4 dx C o.1/w-L1 .dt / C o.1/L1.dt / loc 2 4 dt R3 Z 1 Q 1 C P nN EN C jN ^ BN uQ C j rx nN nQ rx jN dx 2 4 R3 Z 1 ..E ^ B / @t uN .E ˝ E C e C B ˝ B C b / W rx u/ N dx: C 2 R3

376


The next step consists in combining the preceding inequality with the identity (12.66) in order to modulate the Poynting vector E ^ B . This yields 0 Z d @ 1 nQ nN C uQ uN C 1 E EN ^ B BN uN dt R3 4 2 1 1 0 a a 26 35 1 C @a34 a16 A uN A dx 2 a a 15 24 Z Z C Q dvdv d W rx uN dx qO C qO MM C R3 R3 R3 S2 ! N L1 .dx/ k@t uk ıH .t/ C kuk N W 1;1 .dx/ C 1 kuk N L1 .dx/ Z 1 1 1 N N C A4 ^ B B C E E ^ A5 uN A1 uQ nQ rx A4 dx 2 2 4 R3 d C o.1/w-L1 .dt / C o.1/L1.dt / loc dt Z 1 1 uN ^ BN rx nN R;12 R;13 EN uN C nN EN C jN ^ BN P ? uQ dx C 2 R3 2 Z 1 nN EN uQ C nQ EN uN C nN E EN uN dx C 2 R3 Z 1 1 1 N N Q C rx nQ uQ ^ B j C rx nN uN ^ B j dx 2 R3 2 2 Z 1 jN ^ B BN .uN uQ / C E EN ^ B BN @t uN dx C 2 R3 Z 1 E EN ˝ E EN C e C B BN ˝ B BN C b W rx uN dx: 2 R3 It then follows, using the convergence (12.42), the estimates (4.25), (12.63) and Lemma 4.4, that 0 Z d @ 1 nQ nN C uQ uN C 1 E EN ^ B BN uN dt R3 4 2 1 1 0 a a 26 35 1 C @a34 a16 A uN A dx 2 a a 15 24 Z Z C Q dvdv d W rx uN dx qO C qO MM C R3

R3 R3 S2


C

N W 1;1 .dx/ C k@t uk N L1 .dx/ kuk

Z

1 kuk N L1 .dx/

377

! ıH .t/

1 1 1 N N C A4 ^ B B C E E ^ A5 uN A1 uQ nQ rx A4 dx 2 4 R3 2 Z 1 1 d C o.1/w-L1 .dt / C uN ^ BN rx nN R;12 dx o.1/L1.dt / C loc dt 2 R3 2 Z 1 nN EN uQ C nQ EN uN C nN E EN uN dx C 2 R3 Z 1 1 1 N N Q C rx nQ uQ ^ B j C rx nN uN ^ B j dx 2 R3 2 2 Z 1 jN ^ B BN .uN uQ / dx: C 2 R3 Now, for mere convenience of notation, we introduce the following integrand: 1 1 I D nQ nN C uQ uN C E EN C B BN 4 2 1 0 a26 a35 1 1 N E EN ^ B BN uN C @a34 a16 A u: C 2 2 a a 15 24 Thus, further employing the identity (12.65), we find that ! Z Z Z X d Q I dx C qO˙ MM W rx uN dx dvdv d dt R3 R3 R3 R3 S2 ˙ ! Z N W 1;1 .dx/ C k@t uk N L1 .dx/ kuk 1 ıH .t/ C A1 uQ dx 1 kuk N L1 .dx/ 2 R3 Z 1 1 N N A4 E C uN ^ B B rx nQ C A5 B C E E ^ uN dx 2 R3 2 d C o.1/w-L1 .dt / C o.1/L1.dt / loc dt Z 1 1 C rx nQ E uQ ^ B jN nN uN dx 2 R3 2 Z 1 1 N N C rx nN E uN ^ B jQ nQ uQ dx 2 R3 2 Z 1 1 .uQ u/ C N ^ B BN jN nN uN C P ? uQ rx nN 2 dx 2 R3 4 Z 1 1 C ..nQ n/ N .uQ u/ N R;12 / rx nN EN uN ^ BN dx; 2 R3 2

378


whence, in view of the convergence (12.42), the estimate (12.62) and Lemma 12.3 (allowing to control the energy by the entropy), ! Z Z Z X d ˙Q I dx C qO MM dvdv d W rx uN dx dt R3 R3 R3 R3 S2 ˙ N W 1;1 .dx/ C k@t uk N L1 .dx/ C jN nN uN L1 .dx/ kuk C 1 kuk N L1 .dx/ ! Z 1 1 N N C n N E u N ^ B H .t/ A1 uQ dx r ı 2 x 1 2 R3 L .dx/ Z 1 1 A4 E C uN ^ B BN rx nQ C A5 B C E EN ^ uN dx 2 R3 2 d o.1/L1.dt / C o.1/w-L1 .dt / C loc dt Z 1 1 C rx nQ E uQ ^ B jN nN uN dx 2 R3 2 Z 1 1 C rx nN EN uN ^ BN jQ nQ uQ dx: 2 R3 2 Using then the approximate Ohm’s law (12.58) with the control (12.69) and reorganizing the resulting inequality so that all remainder terms appear on its right-hand side, we obtain ! Z Z Z X d Q I dx C qO ˙ MM W rx uN dx dvdv d dt R3 R3 R3 R3 S2 ˙ ! Z Z X 1 Q N N uN dx ˙qO ˙ ˙ qO˙; ˆMM dvdv d j n R3 R3 R3 S2 ˙ N L1 .dx/ C jN nN uN L1 .dx/ N W 1;1 .dx/ C k@t uk kuk C 1 kuk N L1 .dx/ ! 1 d ıH .t/Co.1/w-L1 .dt / C C N EN uN ^ BN o.1/L1.dt / 2 rx n 1 loc dt L .dx/ Z 1 1 N N A4 E C uN ^ B B rx nQ C A5 B C E E ^ uN dx 2 R3 2 " # ! Z Z X ˙ 1 Q A3 C Q dx ˙qO ˙ qO ˙; ˆMM dvdv d A1 u 2 R3 R3 R3 S2 ˙ Z 1 .R;9 C rx R;10 C R;11 / jN nN uN 2 R3 1 rx nN EN uN ^ BN R;14 dx: 2


379

Thus, in view of the estimates (12.59), (12.60), (12.68), (12.70) and Lemma 12.3 (allowing to control the energy by the entropy), we infer that ! Z Z Z X 1 d Q I dx C W rx uN dx qO ˙ C qO˙; MM dvdv d dt R3 2 R3 R3 R3 S2 ˙ ! Z Z X 1 Q N N uN dx ˙qO ˙ ˙ qO˙; ˆMM dvdv d j n R3 R3 R3 S2 ˙ N L1 .dx/ C jN nN uN L1 .dx/ N W 1;1 .dx/ C k@t uk kuk C 1 kuk N L1 .dx/ ! 1 d N N ıH .t/ C o.1/w-L1 .dt / C C o.1/L1.dt / 2 rx nN E uN ^ B 1 loc dt L .dx/ Z 1 1 N N A4 E C uN ^ B B rx nQ C A5 B C E E ^ uN dx 2 R3 2 " # ! Z Z X ˙ 1 Q A3 C Q dx: ˙qO ˙ qO ˙; ˆMM dvdv d A1 u 2 R3 R3 R3 S2 ˙ (12.74) At last, we may now combine the inequalities (12.73) and (12.74) to deduce, employing the symmetries of collision integrands and (12.49) to rewrite dissipation terms, that 0 2 2 Z d @ gC C jvj gN C C g jvj gN C E EN C B BN dt R3 K K 0 11 1 0 a a 26 35 C @ E EN ^ B BN C @a34 a16 AA uN A dx a15 a24 Z C C 1 qO qN C qO qN C qO C; qN C; C qO ;C qN ;C MM dxdvdv d C 2 R3 R3 R3 S2 1 d .t/ıH .t/ C o.1/w-L1 .dt / C o.1/L1.dt / ıD .t/ loc dt 2 0 1 uQ 3 Q B C Q Z 2 B R C

B C C; ;C Q A B R3 R3 S2 qO C qO C qO qO ˆMM dvdv d C dx: B C 3 R @ A E C uN ^ B BN 12 rx nQ N B C E E ^ uN It may be useful to note here that ıH .t/ o.1/L1.dt / , which follows from Lemma 12.3 and implies that C1 ıH .t/ C2 ıH .t/ C o.1/L1.dt / , for any 0 < C1 C2 . Indeed, up to the introduction of vanishing remainders in L1 .dt/, this conveniently allows us to define the coefficient .t/ with one independent constant only.

380


Next, assembling the preceding inequality with the scaled entropy inequality (12.53) and the energy estimate (12.72), we finally obtain d ıH .t/ C ıD .t/ dt

1 d .t/ıH .t/ C o.1/w-L1 .dt / C o.1/L1.dt / C ıD .t/ loc dt 2 Z A C R3 1 0 uQ uN 3 Q C B Q 52 N C B R 2 C B Q N N uN C B R3 R3 S2 qO qO C qO C; qO ;C ˆMM C dx; dvdv d j n C B 1 N N A @ E E C uN ^ B B 2 rx .nQ n/ N N N B B C E E ^ uN

which, with a straightforward application of Grönwall’s lemma (carefully note that this is valid even though ıH .t/ may be negative), concludes the proof of the proposition. Remark. The proof of Proposition 12.4 is based on the construction of the stability inequality (3.22) from Proposition 3.10 for the two-fluid incompressible Navier– Stokes–Maxwell system with Ohm’s law (3.5). As in the proof of Proposition 12.2, this approach has the great advantage of using the approximate macroscopic conservation of momentum established in Proposition 9.6 rather than the one from Proposition 9.5 and, thus, removes the difficulties associated with the nonlinear Lorentz force nQ E C jQ ^ B by expressing it with the Poynting vector E ^ B (and some other terms). However, the drawback of this approach resides in the necessity of the restriction N L1 < 1. Recall, nevertheless, that this restriction is physically relevant, since it kuk t;x merely entails that the modulus of the velocity uN be less than the speed of light (see comments after the proofs of Propositions 3.10 and 3.13). Note finally that it is not possible (at least, we do not know how to make it work) to establish a similar renormalized relative entropy inequality for renormalized solutions of the scaled two-species Vlasov–Maxwell–Boltzmann system (4.35) based on the construction of the stability inequality (3.18) from Proposition 3.8 (see the remark following the proof of Proposition 12.2).

12.3.4 Convergence and conclusion of the proof We may now pass to the limit in the approximate stability inequality (12.52), and thus derive the crucial modulated energy inequality for the limiting system (4.38). To this end, we simply integrate (12.52) in time against non-negative test functions and then


381

let ! 0, which yields, in view of the well-preparedness of the initial data (4.44), the weak convergences (12.37), (12.40), (12.41) and the lower semi-continuities (12.50), (12.51), that Z Rt 1 t ıE .t/ C ıD .s/e s . /d ds 2 0 Rt

ıE .0/e 0 .s/ds Z tZ C A 3 00 R

1 u uN 3 C B 52 N B R C ˙ 2 ˙; P C B Q N ˆMM B R3 R3 S2 ˙ ˙q ˙ q 1 dvdv d j nN uN C .s/ dx A @ N N E E C uN ^ B B 2 rx .n n/ N B BN C E EN ^ uN e

Rt s

. /d

ds:

Further, observe that the characterizations (8.38), (8.39) of the limiting collision integrands q ˙ , q ˙; from Proposition 8.4 imply, using (2.68) and the self-adjointness of L C L, that Z X Q ˙q ˙ ˙ q ˙; ˆMM dvdv d R3 R3 S2 ˙

Z

D R3

Z D

Q dv .L C L/ .h nu ˆ n ‰/ ˆM .h nu ˆ n ‰/ ˆM dv D j nu:

R3

Therefore, using (12.39), we deduce that Z Rt 1 t ıD .s/e s . /d ds ıE .t/ C 2 0 ıE .0/e

Rt 0

.s/ds

1 u uN 5 N C B Z tZ Rt 2 C B C .s/ dxe s . /d ds; N nN uN j nu j C AB C B R3 0 @E EN C uN ^ B BN 1 rx .n n/ N A 2 N N B B C E E ^ uN 0

which is precisely the stability inequality we were after.

382


As for the temporal continuity of u; n; 52 ; E; B , it is readily seen from the approximate macroscopic conservation 3 laws from Proposition 9.5 and Maxwell’s equaQ tions (12.61) that @t P uQ , @t nQ , @t 2 Q , @t E and @t B are uniformly bounded, in L1loc in time and in some negative index Sobolev space in x. It is therefore pos sible to show (see [59, Appendix C]) that P uQ ; nQ ; 32 Q Q ; E ; B converges to u; n; 52 ; E; B 2 C.Œ0; 1/I w-L2 .R3 // weakly in L2 .dx/ uniformly locally in time. At last, the proof of Theorem 4.7 is complete.

Appendices

383

Appendix A

Cross-section for momentum and energy transfer The cross-section for momentum and energy transfer m.z/ D m.jzj/ 2 L1loc .R3 /, such that m.z/ 0, is defined by Z v v 0 b.v v ; /d D m.v v / .v v / ; S2 2 Z 2 jv 0 j2 jv j2 jvj jvj b.v v ; /d D m.v v / : 2 2 2 2 S2 Clearly, it is defined as the average transfer of momentum and energy in any collision between any two particles having pre-collisional velocities v 2 R3 and v 2 R3 . The following proposition guarantees that m.z/ is well-defined by the relations above. Proposition A.1. Let

Z 1 .1 cos / b .jv v j; cos / d; m.v v / D m.jv v j/ D 2 S2 vv with cos D jvv . j It holds that Z v v 0 b.v v ; /d D m.v v / .v v / ; 2 ZS ˇ ˇ ˇv v 0 ˇ2 b.v v ; /d D m.v v / jv v j2 ; 2 Z S

2 jvj2 jv 0 j b.v v ; /d D m.v v / jvj2 jv j2 ; S2

and

Z R3 R3 S2

Z

1 D 3

Z

R3 R3

R3 R3 S2

Z

v v 0 ˝ v v 0 b.v v ; /MM dvdv d

D 0;

R3 R3 S2

m.v v /jv v j2 MM dvdv Id;

2 v v 0 jvj2 jv 0 j b.v v ; /MM dvdv d

2 2 jvj2 jv 0 j b.v v ; /MM dvdv d

Z

D R3 R3

2 m.v v / jvj2 jv j2 MM dvdv :

386

Cross-section for momentum and energy transfer

Proof. Note first that, using the spherical symmetries of the cross-section b, Q b.v v ; / D b.v v ; /; where

v v v v Q D 2 2 S2 : jv v j jv v j

Moreover, the mapping 7! Q preserves the surface measure of the sphere. Hence, we compute Z Z v v jv v j 0 v v b.v v ; /d D b.v v ; /d 2 2 S2 S2 Z C Q v v jv v j D b.v v ; /d 2 2 jv v j S2 Z 1 .1 cos / .v v /b.v v ; /d: D 2 S2 Next, since jv v 0 j2 D 12 .1 cos /jv v j2 , we easily find that Z Z 1 jv v 0 j2 b.v v ; /d D .1 cos / jv v j2 b.v v ; /d: 2 S2 S2 Further straightforward computations yield, employing the previous identities, that Z Z 2 jvj jv 0 j2 b.v v ; /d D 2v .v v 0 / jv v 0 j2 b.v v ; /d S2 S2 D m.v v / 2v .v v / jv v j2 D m.v v / jvj2 jv j2 : Finally, we obtain, using the pre-post-collisional change of variables and the previous identities, Z v v 0 ˝ v v 0 b.v v ; /MM dvdv d R3 R3 S2 Z D2 v v 0 ˝ vb.v v ; /MM dvdv d 3 3 2 ZR R S D2 m.v v / .v v / ˝ vMM dvdv 3 3 Z R R D m.v v /.v v / ˝ .v v /MM dvdv R3 R3 Z 1 D m.v v /jv v j2 MM dvdv Id; 3 R3 R3

Cross-section for momentum and energy transfer

Z

v v0

R3 R3 S2

Z

2 2 jvj jv 0 j b.v v ; /MM dvdv d

D2 ZR

3 R3 S2

D2 Z D

R3 R3

R3 R3

387

v v 0 jvj2 b.v v ; /MM dvdv d

m.v v / .v v / jvj2 MM dvdv

m.v v /.v v / jvj2 jv j2 MM dvdv

D 0; and

Z

jvj2 jv 0 j

R3 R3 S2

Z

D2 R3 R3 S2

Z D2 Z D

R3 R3

R3 R3

2

2

b.v v ; /MM dvdv d

jvj2 jv 0 j

2

jvj2 b.v v ; /MM dvdv d

m.v v / jvj2 jv j2 jvj2 MM dvdv

2 m.v v / jvj2 jv j2 MM dvdv ;

which concludes the justification of the proposition.

Appendix B

Young inequalitites The use of generalized Young inequalities has been ubiquitous in the theory of hydrodynamic limits of the Boltzmann equation since its early treatment in [11]. In its most general version, Young’s inequality (also known as Fenchel–Young inequality or Fenchel’s inequality, in this case) states that hz; yiE;E f .z/ C f .y/;

(B.1)

where E is a real vector space, E is its algebraic dual space, f .z/ is a real-valued functional defined on a domain D E and f .y/ is its Legendre transform (or Legendre–Fenchel transform) defined by (B.2) f .y/ D sup hz; yiE;E f .z/ ; z2D

on the dual domain

D D y 2 E W sup hz; yiE;E f .z/ < 1 :

z2D

Note that D is convex and that f is lower semi-continuous and convex, for it is defined as the supremum of affine functions. Thus, the transform f is also called the convex conjugate of f . Clearly, the inequality (B.1) is an obvious consequence of the definition (B.2). Young inequalities are fundamental in extracting useful information and controls from the entropy and the entropy dissipation bounds in (2.8) and (2.37). To this end, following [11], we introduce the non-negative convex functions h.z/ D .1 C z/ log.1 C z/ z; r.z/ D z log.1 C z/; defined over z > 1. Notice that h.z/ r.z/. In this notation, we may then recast the entropy as Z 1 1 1 H.f / D 2 H.f jM / D h .g / M dxdv; 2 2 R3 R3 where f D M.1 C g /, and the entropy dissipation as 0 0 Z 1 1 f f D.f / D r 1 f f b.v v ; / dvdv d: 4 4 4 R3 R3 S2 f f

390

Young inequalitites

We recall now some useful properties, which are already found in [11], of the convex functions h.z/ and r.z/. Thus, we consider the Legendre transforms h .y/ and r .y/ of h.z/ and r.z/, respectively, well-defined for any y 2 R by h .y/ D ey 1 y; and r .y/ D

z02 ; 1 C z0

z0 . Since h.z/ r.z/, where z0 > 1 is the unique solution to y D log .1 C z0 /C 1Cz 0 notice that h .y/ r .y/. Then, of course, for any z > 1 and y 2 R, the Young inequalities hold:

zy h.z/ C h .y/ D Œ.1 C z/ log.1 C z/ z C Œey 1 y ; zy r.z/ C r .y/ D Œz log.1 C z/ C r .y/: It is also possible to show that r.z/ and h.z/ satisfy the reflection inequalities, for any z > 1, h .jzj/ h.z/; r .jzj/ r.z/; and that h .y/ and r .y/ have the following exponential growth control, for any y 0, h .y/ ey ; r .y/ ey ; and the superquadratic homogeneity, for any y 0 and 0 1, h . y/ 2 h .y/; r . y/ 2 r .y/;

which is easily obtained by proving that 7! h .y/ and 7! r .y/ are increasing 2 2 functions. Thus, combining the above properties we arrive at the most useful inequalities ˛ ˇ 2 jyj ˛ jˇyj 2 h.z/ C e ; jzyˇj 2 h .jzj/ C h ˛ ˛ (B.3) 1 for any z > ; y 2 R; ˛ jˇj > 0; and

2 2 ˛ ˇ 2 jyj jˇyj r jzj C r 4 r 2 z C e ; ˛ ˛ 1 for any z > 2 ; y 2 R; ˛ 2 jˇj > 0:

˛ jzyˇj 4

(B.4)

Young inequalitites

391

The above Young inequalities (B.3) and (B.4) are intensively used throughout this work to extract bounds and compactness properties from the various entropy and entropy dissipation bounds. Finally, for reference, we list some elementary inequalities in connection with the convex functions h.z/ and r.z/ and their corresponding Young inequalities: For every z > 1, it holds that p

2 1 C z 1 h.z/; (B.5) which follows straightforwardly from the identity, for all z > 1,

2 p

p p 1 C z 1 C 2h 1Cz1 1 C z D h.z/; or by showing that the function defined on z > 1 by ( h.z/ if z ¤ 0; p 2 f .z/ D . 1Cz1/ 2 if z D 0; is increasing and reaches the value 1 as z ! 1. For every z > 1, it holds that 1 1 2 z h.z/ C z 3 ; 2 6 which is obtained by integrating twice the elementary inequality 1

(B.6)

1 C z D h00 .z/ C z; 1Cz

valid for every z > 1. For every z > 1, it holds that p

1Cz1

2

jzj;

(B.7)

which is a consequence of the direct computation ( p

2 p 2 1Cz1 if z 0; p p 1Cz1 D jzj 2 1 C z 1 1 C z if z 0: For every z > 1, it holds that

2 1 p 1 C z 1 r.z/; 4 which follows straightforwardly by integrating twice the inequality 1 1 1 1 : C 3 2 1Cz .1 C z/2 .1 C z/ 2

(B.8)

392

Young inequalitites

For every z > 1 and y 2 R, it holds that zy D zy1fy0;ylog.1Cz/g[fylog.1Cz/g C zy1fy0;z<ey 1g[fy0 is not known a priori and when each is non-negative, based on the proof of Theorem 2.4 from [7]. Recall that, according to the partial proof following the statement of Lemma 7.7, it is sufficient in this case to establish the equiintegrability of f .t; x; v/g>0 in all variables. For the reader’s convenience, we first recall the precise result which we are about to justify. Lemma C.1. Let the bounded family of non-negative functions f .t; x; v/g>0 L1 Rt R3x I Lr R3v ; for some 1 < r < 1, be locally relatively compact in v and such that ˇ

˛

.@t C v rx / D .1 x / 2 .1 v / 2 S ; for all > 0 and for some bounded family fS .t; x; v/g>0 L1 Rt R3x I Lr .R3v / ; where ˛ 0 and 0 ˇ < 1. We further assume that, for any compact set K R3 R3 , Z .t; x; v/ dxdv is equi-integrable (in t). K

>0

Then, f .t; x; v/g>0 is equi-integrable (in all variables). We advise the reader of the difficulty to grasp the full content of the proof below without any prior knowledge of the work from [7]. Therefore, we suggest that this appendix be read in parallel with the article [7]. Proof. We first notice, repeating the proof of Lemma 3.1 from [7], that we have the following interpolation formula (compare with (5.17) in [7]), for any R > 0 and ı > 0, denoting by .; ; / the Fourier variables of .t; x; v/ and by F the Fourier

394

End of proof of Lemma 7.7 on hypoelliptic transfer of compactness

transforms: 1 Fx;v .1 / Fx;v .t; x; v/ R

Z 1

3 O . .u v// K1 .t s; x y; u/ D .2/3 RR3 R3 1 Fx;v .1 / Fx;v .s; y; u/ dsdydu R

Z 2ıi

3 O . .u v// K2 .t s; x y; u/ .2/3 RR3 R3 " # ˇ hi˛ 1 1ˇ hi R Fx;v .1 / Fx;v S .s; y; u/ dsdydu: R jj 10 ˛ (C.1) 1ˇ 1 1 3 Here D ı 1C˛ R 1C˛ , ; 2 Cc .R / are cutoff functions such that 1fjrj 1 g 2 .r/ 1fjrj1g and 1f1jrj5g .r/ 1f 1 jrj 11 g , and the singular kernels Ki , 2 2 i D 1; 2, are defined by 2

1 . u C / ; K1 .t; x; u/ D Ft;x jj 2

1 K2 .t; x; u/ D Ft;x . u C / ; jj R 1 O /d D 1 and supp O f1 j j 2g, where 2 S .R/ is such that .0/ D 2 R . 1.r/ and .r/ D r for all r 2 R. The rather deep meaning of the above interpolation (C.1) resides in the formula presence of the frequency cutoff functions and in its right-hand side, 10 which is only possible through a precise analysis of dispersive and hypoelliptic phenomena, i.e., the transport of frequencies, in the kinetic transport equation. This requires that the support of . O / be restricted to f1 j j 2g, though. Now, we claim that the mappings Z 1 f .t; x; v/ 7!

3 O . .u v// Ki .t s; x y; u/f .s; y; u/ dsdydu; 3 3 3 .2/ RR R (C.2) with i D 1; 2, have bounded extensions between (C.3) L1 Rt R3x I Lr R3v ! Lr R3v I L1;1 Rt R3x :


395

As shown in [7], this boundedness follows from an application of Theorem 5.2 therein (or a very slight variant of it allowing different dimensions for different variables, i.e., .t; x/ 2 R4 and v 2 R3 ), provided we establish that Z sup sup v2R3 .s;y/¤0 fj.t;x/j2j.s;y/jg < 1; jAv ]Ki .t s; x y; u/ Av ]Ki .t; x; u/j dtdx r0 3 L .j ..uv// O jdu/ (C.4) where Av ]Ki .t; x; u/ D jdet Av j Ki .Av .t; x/; u/, for some family of automorphisms Av of R R3 fixing the origin (the remaining hypotheses of Theorem 5.2 from [7] being easily verified through standard arguments from the analysis of Fourier multipliers). Here, we will consider, for each v 2 R3 , the automorphism Av of R R3 defined by Av .t; x/ D . t; x C tv/ ; so that

.u v/ Av ]K1 .t; x; u/ D 2

C2 ; jj jj .u v/ 1 2

Av ]K2 .t; x; u/ D Ft;x C2 : jj jj 1 Ft;x

In turn, the estimate (C.4) is established employing classical methods from harmonic analysis (in [7], the corresponding step is performed in Lemmas 5.3 and 5.4). Thus, it can be shown that (C.4) will hold as soon as the following Hörmander– Mikhlin condition for homogeneous Fourier multipliers is verified: Z ˇ ˇ ˇ@ Ft;x .Av ]Ki / .; ; u/ˇ2 d d C.1 C ju vj/N ; (C.5) ; 1<j.; /j 0 and some possibly very large N 2 N, and for any multi-index 2 N4 such that 1 2 f0; 1g and 0 2 C 3 C 4 2. The preceding control is easily verified for K1 through a straightforward calculation using that .r/ decays rapidly. As for K2 , this step requires some greater care 1 for large values of r, for all n 2 N. Neverbecause .n/ .r/ only decays as jrjnC1 theless, it is also straightforwardly verified upon noticing, for any multi-index as

396


before, that

ˇ ˇ ˇ ˇ .u v/ ˇ C2 j.; /j ˇˇ@; 2

jj jj ˇ ˇ ˇ 3 X j j nC2 ˇˇ .n/ .u v/ ˇˇ C 2

.1 C ju vj/n 1 C C 2 ˇ jj jj jj ˇ nD0 3 j j X .1 C ju vj/2nC1 C 1C jj nD0 ˇ ˇ ˇnC1 ˇˇ ˇ ˇ .u v/ ˇ .u v/ ˇ ˇ ˇ ˇ 1 C ˇˇ

.n/ 2

C C2 ˇ ˇ ˇ jj jj jj jj ˇ C j j .1 C ju vj/7 .1 C ju vj/7 ; C 1C jj jj jj

which implies that (C.5) holds with N D 14, for j j12 is locally integrable in R3 . So far, we have thus established the weak type boundedness on the spaces (C.3) of the mappings (C.2). Therefore, we conclude from the interpolation formula (C.1) that h i 1 1 Fx r 1;1 C Fv .1 / Fv Fx .1 / 1 r R

Lv Lt;x Lt;x Lv 1 CC Fv Fv 1 r

Lt;x Lv C C ı kS kL1

r t;x Lv

;

whence, since the ’s are relatively compact in v, since limR!1 D 1 and by the arbitrariness of ı > 0, h i lim sup Fx1 .1 / Fx r 1;1 D 0: (C.6) R!1 >0 R Lv Lt;x

i Next, applying spatial Riesz transforms f 7! Fx1 j j Fx f , i D 1; 2; 3, to the identity (C.1) and repeating the preceding arguments, we deduce that it also holds 1 i Fx (C.7) lim sup Fx .1 / r 1;1 D 0: R!1 >0 R jj Lv Lt;x

Consequently, we obtain that the ’s are relatively compact in x in the quasi-Banach space defined by the quasi-norm kf kLr L1;1 C v

t;x

3 X 1 i F F f x jj x i D1

Lrv L1;1 t;x

;


397

which can be shown, repeating standard (but difficult) arguments from the analysis of weak Hardy spaces (see [7] and the references therein for details), to be equivalent to the quasi-norm sup j's x f j ; (C.8) Lrv L1;1 t;x

s>0

R

where '.x/ 2 Cc1 .R3 / is such that R3 '.x/dx ¤ 0 and 's .x/ D s13 '. xs /. Finally, following [7], we explain how the ensuing spatial compactness in the topology given by the quasi-norm (C.8) is sufficient to entail the equi-integrability of the ’s, provided they are non-negative. To this end, we note first, for every a; b > 0 and any compact subset K R R3 R3 , that Z j .t; x; v/j dtdxdv K\fj j>ag Z j .t; x; v/j dtdxdv K\ j j>a; kf k

L1 x

Z

C

k k

>b L1 x

b

k1K .t; x; v/kL1x dtdv:

Clearly, in view of the equi-integrability hypotheses, the family k .t; x; v/kL1x is locally equi-integrable in t and v, so that the last term above can be made uniformly small by choosing b > 0 arbitrarily large. Thus, the proof will be finished upon showing that, for each fixed b > 0 and every compact subset K R R3 R3 , Z j .t; x; v/j dtdxdv D 0; lim sup a!1 >0

.t;x;v/2K W j j>a; k k

L1 x

b

which, following the arguments from the end of the proof of Theorem 2.4 in [7] based on Proposition 5.5 therein (which is only valid for non-negative functions), is a direct consequence of the compactness estimates (C.6) and (C.7). The justification of the lemma is thus complete.

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Index acceleration operator, 152, 343, 347, 366 acceleration terms, 244, 265, 276, 280, 284, 285, 293 acoustic waves, 25, 26, 55, 313, 318 acoustic-electromagnetic waves, 145, 320 adjusted electric field, 328 admissible renormalization, 123 advection term, 31, 273 Ampère’s equation, 3, 9, 151, 198, 334 approximate conservation laws, 263, 265, 278 approximate conservation of momentum, 283, 349 approximate macroscopic conservation laws, 323, 338, 363 approximate Ohm’s law, 352 asymptotic systems, 31, 33, 52, 65, 67 Bessel inequality, 191, 260 Boltzmann collision operator, 3, 5 Boussinesq relation, 26, 56, 149 bulk velocity, 11, 41, 263 collision integrands, 253 collision invariant, 8, 22 collision kernel, 5, 6 compactness of the gain term, 200 compensated compactness, 81, 208, 246, 252, 318 conservation defects, 14, 233, 237, 264– 266, 280, 285, 287 conservation laws, 9, 12 consistency estimate, 284, 286 continuity equation, 229 Coulomb gauge, 29, 82, 326 cross-section, 6 cross-section for momentum and energy transfer, 63, 385 cutoff assumption, 6 cutoff collision kernels, 14

damped wave equation, 82 defects in macroscopic conservation laws, 139 diffusion term, 31 dissipative solution, 59, 73, 84, 89, 95, 147, 331, 335 Dunford–Pettis compactness criterion, 126 Egorov’s theorem, 307 electric charge, 43 electric conductivity, 234, 257, 258, 334 electric current, 43, 234, 313 electric repulsion, 19 electrodynamic continuity equation, 229 electrodynamic pressure, 64 electrodynamic variables, 43, 44, 46, 150, 154, 313 electromagnetic energy flux, 80 electromagnetic waves, 25, 26, 55, 313, 318 energy conductivity, 258 energy equivalence relation, 61 energy inequalities, 199, 258 entropic convergence, 143, 149, 153 entropy, 5, 10 entropy dissipation, 10, 11 entropy dissipation bound, 161, 268 entropy inequality, 12, 123, 141, 146 equi-integrability, 213, 266, 270, 277 equi-integrability in v, 199 Faraday’s equation, 3, 9, 53, 68, 246, 252 fast relaxation limit, 15 fast time-oscillations, 25, 313 filtering method, 313 flux terms, 29, 242, 270, 290 fluxes, 265, 279, 285 gain term, 6 gauge transformation, 82

404

Index

Gauss’ laws, 3, 9 Gaussian decay estimate, 276 Gaussian distributions, 275 Gaussian tails, 243, 296, 300 generalized relative entropy method, 145, 198 global conservation of energy, 136 global normalized Maxwellian equilibrium, 122 Grad’s moment method, 323 growth rate, 367 H -theorem, 11, 12 hard potentials, 8 Hilbert expansions, 331 hydrodynamic projection, 23, 41 hydrodynamic variables, 44, 150, 154, 282 hypoelliptic transfer of compactness, 14, 213, 393 hypoellipticity, 199, 208 ill-prepared initial data, 155 impact parameter, 7 improved velocity integrability, 174, 286 incompressibility relation, 26, 56, 149 incompressible Navier–Stokes–Fourier system, 33, 67 incompressible Navier–Stokes–Fourier system in a Boussinesq regime, 36 incompressible Navier–Stokes–Fourier– Maxwell system, 34 incompressible Navier–Stokes–Fourier– Poisson system, 35, 37 incompressible Navier–Stokes–Maxwell system, 81 incompressible quasi-static Navier– Stokes–Fourier–Maxwell system, 34 incompressible quasi-static Navier– Stokes–Fourier–Maxwell– Poisson system, 35, 73 infinitesimal Maxwellian, 22, 58 internal electric energy, 43, 234

internal electric energy constraint, 234, 256, 334 inverse power potential, 7 kinetic energy flux, 24 kinetic momentum flux, 24 Knudsen number, 15 L1 mixing lemma, 209 Legendre transforms h .y/ and r .y/, 390 Leray projector, 27 Leray solution, 75 limiting energy inequality, 258 linearized Boltzmann operator L, 24, 164 linearized collision operator, 164–166 linearized version of the Chapman– Enskog decomposition, 270 local conservation laws, 8 local conservations of momentum, 136 locally relatively compact in v, 208–210 long-range interaction, 6, 199 Lorentz force, 3, 19, 80, 99, 104, 208, 282, 355, 380 Lorenz gauge, 82 loss term, 6 Mach number, 15 macroscopic conservation laws, 134, 282 macroscopic constraints, 25 macroscopic density, 11 macroscopic fluctuations, 23 macroscopic fluctuations of density, 41 macroscopic hydrodynamic variables, 43 magnetic induction, 19 magnetohydrodynamic system, 69 Maxwell’s equations, 3, 9, 72, 80 Maxwellian distribution, 10, 11 Maxwellian molecules, 8 mean free path, 15 measure-valued renormalized solutions, 13, 14 mixed collision terms, 284 mixed entropy dissipation, 10

Index

modulated dissipation, 99 modulated energies, 337 modulated energy, 99, 111, 114, 148, 152 modulated energy dissipation, 106, 111, 114, 148, 152 modulated energy inequality, 102, 116, 380 modulated entropy, 278 modulated entropy dissipation, 89 modulated Lorentz force, 101, 115 modulated Poynting vector, 101, 115 moment method, 5

405

renormalized modulated entropy inequality, 335, 342 renormalized modulated entropy method, 282, 284, 314 renormalized solution, 5, 13, 122, 123 renormalized solution of the Vlasov– Boltzmann equation, 123 renormalized solutions with a defect measure, 14 Reynolds number, 15

scaled entropy inequality, 17, 18, 22, 36, 70, 323 scaled modulated entropy, 332 Navier–Stokes–Fourier system in a soft potentials, 8 Boussinesq regime, 71 solenoidal Ohm’s law, 68, 108, 114, 147, non-cutoff, 7 234, 263, 334 nonlinear constraint equations, 46 speed of light, 4, 380 nonlinear weak compactness, 183, 219 normalized Maxwellian equilibrium, 16 speed of sound, 15 stability inequality, 151, 332 Ohm’s law, 46, 61, 80, 89, 147, 183, 256, strong Boussinesq relation, 27, 49 strong interspecies interactions, 17, 39, 331 44 Poynting vector, 80, 99, 104, 282, 350, Strouhal number, 15 355, 376, 380 Product Limit Theorem, 230, 269, 270, tails of Gaussian distributions, 238 273, 274, 277, 326, 360 temperature, 11, 41, 263 temporal continuity, 328 quasi-static approximation, 197 thermal conductivity, 258 thermodynamic equilibrium, 11, 164 Radon measures, 135, 137 two-fluid incompressible Navier–Stokes– relative entropy, 11, 158 Fourier system, 52 relative entropy inequality, 11 two-fluid incompressible Navier–Stokes– relative entropy method, 5, 59, 85, 99, Fourier system in a Boussinesq 331 regime, 70 relaxation estimate, 168, 170 two-fluid incompressible Navier–Stokes– renormalized collision integrands, 162 Fourier–Maxwell system, 52 renormalized collision operator, 124 two-fluid incompressible Navier–Stokes– renormalized fluctuations, 160 Fourier–Maxwell system with Ohm’s law, 66, 73 renormalized modulated energy, 339, 364 two-fluid incompressible Navier–Stokes– Fourier–Maxwell system with renormalized modulated entropy, 338, solenoidal Ohm’s law, 68, 73 339, 362, 363 renormalized modulated entropy dissipa- two-fluid incompressible Navier–Stokes– Fourier–Poisson system, 55, 71 tion, 339, 364

406

Index

vacuum permittivity, 4 vectorial linearized Boltzmann operators, 40 velocity averaging lemma, 127 very weak interspecies interactions, 17, 39, 44 viscosity, 258 Vlasov–Boltzmann equation, 3, 122 Vlasov–Maxwell–Boltzmann system, 3, 36 Vlasov–Poisson–Boltzmann system, 36 von Kármán relation, 15

two-fluid incompressible Navier–Stokes– Fourier–Poisson system with Ohm’s law, 66, 71 two-fluid incompressible quasi-static Navier–Stokes–Fourier– Maxwell system, 53 two-fluid incompressible quasi-static Navier–Stokes–Fourier– Maxwell system with solenoidal Ohm’s law, 68 two-fluid incompressible quasi-static Navier–Stokes–Fourier– Maxwell–Poisson system, 54 two-fluid incompressible resistive Navier–Stokes–Fourier system, 66 two-fluid incompressible resistive Navier–Stokes–Fourier system in a Boussinesq regime, 71 two-species Vlasov–Poisson–Boltzmann system, 69

wave operator, 25, 314, 321 weak Boussinesq relation, 27, 49 weak interspecies interactions, 17, 39, 44 weak solution, 73, 75 weak-strong stability, 84, 88, 95, 108, 331, 335 weak-strong uniqueness principle, 76 well-prepared initial data, 149, 153, 154, 314, 331

vacuum permeability, 4

Young inequalitites, 389

Volume 1 The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subject to self-induced electromagnetic forces. At the macroscopic scale, in the incompressible viscous fluid limit the evolution of the plasma is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing that may take on various forms depending on the number of species and on the strength of the interactions. From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way. The first part of this work is devoted to the systematic formal analysis of viscous hydro dynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature. In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman– Enskog-type decomposition and regularity in the v variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more). The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known.

ISBN 978-3-03719-193-4

www.ems-ph.org

Monographs / Arsénio/Saint-Raymond Vol. 1 | Font: Rotis Semi Sans | Farben: Pantone 116, Pantone 287 | RB 24 mm

Diogo Arsénio and Laure Saint-Raymond

From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics

From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics. Volume 1


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