Proceedings
"WASCOM 2005" 13th Conference on
Waves and Stability in Continuous Media
UV Editors
Roberto Monaco Giuseppe Mulone Salvatore Rionero Tommaso Ruggeri World Scientific
Proceedings
"WASCOM 2005" 13th Conference on
Waves and Stability in Continuous Media
This page is intentionally left blank
Proceedings
"WASCOM 2005" 13th Conference on
Waves and Stability in Continuous Media Catania, Italy 19-25 June 2005
Editors
Roberto Monaco Politecnico di Torino, Italy
Giuseppe Mulone Universitd di Catania, Italy
Salvatore Rionero Universitd di Napoli, Italy
Tommaso Ruggeri Universitd di Bologna, Italy
YJ? World Scientific N E W JERSEY
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WAVES AND STABILITY IN CONTINUOUS MEDIA Proceedings of the 13th Conference on WASCOM 2005 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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PREFACE
Since its first edition organized in 1981, the Meeting "International Conference on Waves and Stability in Continuous Media (WASCOM)" is aimed at bringing together Italian and foreign researchers who are interested in stability and wave propagation problems in continuous media, and to discuss leading aspects of these areas of research. This meeting has taken place, since then, every two years with increasing interest and participation. The latest conference, the XIII edition, was held in Acireale (Catania), more precisely in the village of Santa Tecla, June 1925, 2005. Previous Conferences were held in Catania (1981), Arcavacata di Rende (Cosenza, 1983), Giovinazzo (Bari, 1985), Taormina (Messina, 1987), Sorrento (Napoli, 1989), Acireale (Catania, 1991), Bologna (1993), Altavilla Milicia (Palermo, 1995), Capitolo di Monopoli (Bari, 1997), Vulcano (Messina, 1999), Porto Ercole (Grosseto, 2001) and Villasimius (Cagliari, 2003). After each edition, a volume of proceedings documenting the research work and progress in the area is published. The research groups promoting the XIII edition of the conference belong to the Departments of Mathematics of the universities of Bologna, Lecce, Messina, Napoli, Palermo, Torino-Politecnico and, of course, Catania which was in charge of organizing the meeting. These groups, since many years, are very active in the field of waves and stability and belong to a national project supported by the Minister of Education. The XIII edition registered over one hundred participants coming from more than 15 different countries. A special session was dedicated to Prof. Dionigi Galletto (University of Turin) and to Prof. Ingo Miiller (Technical University of Berlin) who have participated to several editions of the conference and that have given important scientific contributions in their respective research areas.
v
VI
T h e topics covered were • • • • • • •
Non-linear wave propagation Linear and non-linear stability in fluid dynamics Small parameter problems Kinetic theories towards continuum models Extended thermodynamics Transformation groups and reduction techniques Numerical and technical applications.
T h e meeting encompassed various sectors of waves and stability such as global existence of smooth solutions and Riemann problem for hyperbolic systems of balance laws and related wave phenomena, diffusion in biology, diffusion in continuum mechanics, with applications in the environment and technology as, for example, pollution control. Non-linear stability and waves featured strongly, so a wide relevance has been devoted to finite time blow-up in P D E ' s , flows in porous media, low t e m p e r a t u r e phenomena such as second sound waves, kinetic models, extended thermodynamics, group analysis techniques t h a t allow to perform an analysis more deepened of the proposed models, regarding b o t h the constitutive aspects (techniques of classification) and the search of exact solutions. This volume contains 73 papers which have been presented at the XIII Conference as invited lectures and short communications. T h e Editors of this volume would like to t h a n k the Scientific Committee who carefully suggested the invited lectures and selected the contributed papers, as well as the members of the Organizing Committee, coming from the Department of Mathematics and Informatics of the University of Catania. A t h a n k also to Sandra Pieraccini (Politecnico di Torino) who has carefully prepared the final editing of the manuscript. A special thank is addressed to all the participants to whom ultimately the success of the conference has to be ascribed. Finally, the Editors are especially indebted to Fondazione CRT di Torino which has partially supported the publishing expenses of the present book. J a n u a r y 2006
T h e Editors Roberto Giuseppe Salvatore Tommaso
Monaco Mulone Rionero Ruggeri
C O N F E R E N C E DATA WASCOM 2005 13th International Conference on Waves and Stability in Continuous Media Acireale (Catania), Italy, June 19-25, 2005 Scientific Committee Chairmen: G. Mulone (Catania), S. Rionero (Napoli) and T. Ruggeri (Bologna) C. Dafermos (Providence), L. Desvillettes (Cachan), J. Flavin (Galway), D . Fusco (Messina), H. Gouin (Marseille), A. M. Greco (Palermo), R. Monaco (Torino), I. Miiller (Berlin), B. Straughan (Durham), C. Tebaldi (Torino)
Organizing Committe Chairmen: M. Torrisi (Catania) and A. Valenti (Catania) P. Carbonaro (Catania), S. Lombardo (Catania), G. Mulone (Catania), O. Muscato (Catania), V. Romano (Catania), R. Tracina (Catania), M . Trovato (Catania) Supported by • Research Project of National Interest (MIUR-COFIN 2003/05) "Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media" (National Coordinator Prof. T. Ruggeri) • Gruppo Nazionale per la Fisica Matematica - INDAM • Universita degli Studi di Catania • Facolta di Scienze MM.FF.NN. di Catania • Facolta di Ingegneria di Catania • Dipartimento di Matematica e Informatica di Catania • Provincia Regionale di Catania • Comune di Acireale • Comune di Riposto • ERSU di Catania • Azienda Provinciale di Turismo di Catania VII
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CONTENTS
Preface Conference Data S. Abenda, T. Grava, G. Klein On Whitham Equations for Camassa-Holm
1
G. AH, I. Torcicollo, S. Vessella Inverse Doping Problems for Semiconductor Devices
7
F. Ancona, S. Bianchini Vanishing Viscosity Solutions of Hyperbolic Systems of Conservation Laws with Boundary
13
F. Bagarello An Operatorial Description of Stock Markets
22
C. Baranger, S. Pieraccini Numerical Simulation of Models for Reacting Polytropic Gases
28
E. Barbera, I. Miiller Heat Conduction in a Non-Inertial Frame
35
M. V. Bartuccelli Nonlinear Stability of Uniform Steady States and Preservation of Sign for a Two-Dimensional Fourth-Order Reaction Diffusion Equation 41 F. Borghero, F. Demontis, S. Pennisi The Non-Relativistic Limit of Relativistic Extended Thermodynamics with Many Moments. Part I: The Balance Equations 47
IX
G. Borgioli, G. Frosali, C. Manzini Hydrodynamic Models for a Two-Band Nonzero-Temperature Quantum Fluid
53
R. E. Caflisch, M. Sammartino Vortex Layers in the Small Viscosity Limit
59
P. Capodanno, D. Vivona Mathematical Study of the Small Oscillations of a Pendulum Filled by an Inviscid, Incompressible, Almost Homogeneous Liquid
71
F. Capone, M. Gentile, S. Rionero Influence of Linear Concentration Heat Source and Parabolic Density on Penetrative Convection Onset
77
F. Capone, M. Gentile, S. Rionero On Penetrative Convection in Porous Media Driven by Quadratic Sources
83
P. Carbonaro Non-linear Schrodinger Equation in a Two-Fluid Plasma
89
M. C. Carrisi, F. Demontis, S. Pennisi The Non-Relativistic Limit of Relativistic Extended Thermodynamics with Many Moments. Part II: How it Includes the Mass, Momentum and Energy Conservation 95 C. Cercignani Existence and Energy Conservation for the Boltzmann Equation
101
Y. Choquet-Bruhat From the Big Bang to Future Complete Cosmologies
110
V. A. Cimmelli, P. Van First Order Weak Nonlocality in Extended Thermodynamics of Rigid Heat Conductors 122
XI
G. M. Coclite, K. H. Karlsen A Semigroup of Solutions for the Degasperis-Procesi Equation
128
V. Colombo, E. Ghedini, A. Mentrelli Numerical Simulation of Magnetically Deflected Transferred Arc
134
F. Conforto, M. Groppi A Note on Balance Laws for Slow and Fast Chemical Reactions
140
R. Conte Integration of Partially Integrable Equations
146
F. Crispo On the Stability and Asymptotic Stability of Steady Solutions of the Navier-Stokes Equations in Unbounded Domains 158 C. Curro, G. Volenti, M. Sugiyama Reflection and Transmission of Acceleration Waves in Isotropic Solids when a Strong Discontinuity Occurs 164 F. Dell'Isola, N. Ianiro, L. Placidi Instability of a Pre-Stressed Solid-Fluid Mixture
170
F. Demontis, S. Pennisi, F. Rundo Some Further Considerations on the Galilean Relativity Principle in Extended Thermodynamics
176
M. Destrade, G. Saccomandi Waves and Vibrations in a Solid of Second Grade
182
L. Desvillettes About the Large Time Behavior of Dissipative Equations when a priori Bounds are Slowly Growing
193
J. Engelbrecht, A. Berezovski, F. Pastrone, M. Braun Deformation Waves in Microstructured Solids and Dispersion
204
Xll
P. Fergola, M. Cerasuolo An Allelopathic Competition with an External Toxicant Input
210
J. N. Flavin, M. F. McCarthy, S. Rionero Stability and Other Considerations for a Nonlinear Diffusion System
220
D. Fusco, N. Manganaro Generalized Rarefaction Waves and Riemann Problem for a Class of Dissipative Hyperbolic Models
232
G. Gambino, M. C. Lombardo, M. Sammartino An Equilibrium Point Regularization for the Chen System
244
M. L. Gandarias, S. Saez Nonclassical Symmetry Reductions of the Calogero-Degasperis-Fokas Equation in (2+1) Dimensions 250 A. Georgescu, A. Labianca, L. Palest A Linear Instability Analysis of the Benard Problem for Deep Convection
256
V. Giovangigli Multicomponent Reactive Flows
262
H. Gouin Non-linear Waves in Fluids Near the Critical Point
274
M. Groppi, G. Spiga On Euler Closures for Reactive Boltzmann Equations
286
R. Kaiser On the Geomagnetic Direction Problem: A Nonexistence Result
292
B. G. Konopelchenko, A. Moro On the Models of Nonlocal Nonlinear Optics
298
Xlll
G. M. Kremer, M. Pandolfi Bianchi, A. J. Soares Closure of the Balance Laws for Gaseous Mixtures Near Chemical Equilibrium
304
D. Lacitignola, C. Tebaldi Caothic Patterns in Lotka-Volterra Systems with Behavioral Adaptation
310
S. La Rosa, V. Romano An Euler-Poisson Model Based on MEP for Holes in Semiconductors 316 S. F. Liotta, G. Mascali A Hydrodynamical Model for Silicon Bipolar Devices
322
M. Lisi, S. Totaro Analysis of a Mathematical Model for the Interaction between Algae and Light
328
G. Lo Bosco, M. Sammartino, V. Sciacca Singularities for Prandtl's Equations
334
M. C. Lombardo, M. Sammartino Nonlocal Boundary Conditions for the Navier-Stokes Equations
340
J. Lou, T. Ruggeri On the Shizuta-Kawashima Coupling Condition for Dissipative Hyperbolic Systems and Acceleration Waves
346
P. Maremonti Pointwise Stability of Solutions of the Navier-Stokes Equations
356
L. Margheriti, C. Tebaldi Bifurcation Analysis for Symmetric Equilibria with Localised Magnetic Shear in 2D RRMHD
365
R. Monaco, M. Pandolfi Bianchi, A. J. Soares Simulations at Kinetic Scale of Relaxation Models for Slow and Fast Chemical Reactions
378
XIV
G. Mulone An Operative Method to Define Generalized-Energy Functional in PDEs and in Convection Problems
390
0. Muscato On the Existence of High Energy Tails for the Boltzmann Transport Equation in Semiconductors
402
F. Oliveira Stability of Solitons of the Zakharov-Rubenchik Equation
408
F. Oliveira, A. J. Soares Global Solutions of Boltzmann-type Equations with Three-Body Chemical Interactions
414
F. Oliveri, G. Manno, R. Vitolo On an Inverse Problem in Group Analysis of PDE's: Lie-Remarkable Equations 420 L. Palese On the Stability of the Magnetic Anisotropic Benard Problem with Hall and Ion-Slip Currents
432
D. F. Parker Higher-Order Shallow Water Equations, Explicit Solutions and the Camassa-Holm Equation
438
S. Pennisi, M. C. Carrisi, A. Scanu The Galilean Relativity Principle as Non-Relativistic Limit of Einstein's One in Extended Thermodynamics
448
S. Pennisi, M. C. Carrisi, A. Scanu Equivalence of Two Known Approaches to Extended Thermodynamics with 13 Moments 455
XV
S. Rionero Functionate for the Coincidence between Linear and Nonlinear Stability with Applications to Spatial Ecology and Double Diffusive Convection
461
V. Romano, M. Torrisi, R. Tracina Symmetry Analysis for the Quantum Drift-Diffusion Model of Semiconductors
475
M. Ruggieri, A. Volenti Symmetries and Reduction Techniques for a Dissipative Model
481
M. Senthilvelan, M. Torrisi Symmetry Analysis and Linearization of the (2+1) Dimensional Burgers Equation
493
M. P. Speciale, M. Brocato Transfer Properties of Elastic Materials with Thin Layers
505
M. Sugiyama Wave Propagation Phenomena in Solids Near the Melting Point
512
M. Svanadze Plain Waves and Vibrations in the Elastic Mixtures
524
M. Trovato Maximum Entropy Principle for Hydrodynamic Analysis of the Fluctuations of Moments for the Hot Carriers in Semiconductors
530
V. Vasumathi, M. Daniel Soliton Excitations in an Inhomogeneous DNA Molecular Chain
536
W. Wang Stability of Structured Prey-Predator Model
542
M. Webber, B. Straughan Decay Estimates in Chemotaxis: Aggregation of Glia and a Possible Application to Alzheimer's Disease Senile Plaques
548
XVI
W. Weiss Extended Thermodynamics with Consistent Order
553
N. Zhao, M. Sugiyama One-Dimensional Stationary Heat Conduction in a Rarefied Gas at Rest Analyzed by Consistent-Order Extended Thermodynamics
559
ON WHITHAM EQUATIONS FOR
CAMASSA-HOLM
S. A B E N D A Department
of Mathematics and C.I.R.A.M. Via Saragozza 8, E-mail:
Research Center of Applied Mathematics University of Bologna, 1-40123 Bologna BO, Raly
[email protected] -
T . GRAVA S.I.S.S.A., Via Beiruth, 9 1-40100 Trieste, Raly E-mail:
[email protected] A N D C. K L E I N Max Planck
Institute for Mathematics in D-0410S Leipzig, Germany E-mail:
[email protected] Science,
The solution of the Cauchy problem for the Camassa-Holm equation tpt + 3(ptpx = £2( = (/>(x, t). The
*INFN-Gruppo c. Cosenza tWork partially supported by GNFM "Problemi di diffusione e controllo di inquinanti nei fluidi e nei porosi" and by PRIN "Stabilita in energia nei continui dissipativi".
7
8
variables satisfy the following scaled drift-diffusion system 9 ,
'-A 2 div
(eW(p)=C+p-n,
< -g£+divJn=J?, dv -£+divJp = -R, K at
(!)
with (x, t) € £1 x (0, T). We introduce the electron and hole Slotboom variables u, v defined by n — 52e^u, p — d2e~^v, where S2 is the scaled intrinsic concentration. Then, the electron and hole flux densities (currents) J „ , J p , are given by J n = (52/xne^Vw, J p = —52fj,pe^^Vv. The other quantities appearing in (1) are the scaled Debye length A, the dielectric constant e = e(x), the electron and hole mobilities fin = /i n (x, n,p, V), fip = A*p(x, n,p, V) and the recombination-generation term R = 54F(x, n,p, V(f>) (uv — 1). We assume that the boundary is made of a Dirichlet part (Ohmic contacts) and a Neumann part (insulating parts), dfl = YD U TJV, YD Pi Tjv = 0, and assign initial-boundary conditions — In u = In v = 4> — 4>h\ —
VD
v • Vu = v • Vv = v • V X — ±b(t), Xb{t) being the position of the boundary. Note that we do not make any assumption on the relations between the characteristic speeds of A(t, u) and the speed of the boundary. In particular the boundary may have the same speed of a characteristic family (boundary characteristic) and this family may change in time. An existence result for hyperbolic boundary value problems was proved in 18 using an adaptation of the Glimm scheme introduced in 17 . Improvements of Goodman's result 18 have been obtained by a wave-front tracking
13
14 technique introduced in 7 and later used in a series of papers MO,11,15,13,12,14 to establish the well posedness of the Cauchy problem. Such a wave-front tracking technique was adapted to the initial-boundary value problem in 1 , where a substantial improvement of Goodman's results 18 was achieved. The well posedness of the initial-boundary value problem was then proved in 16 relying on the wave-front tracking technique described in 1 . All the results quoted so far deal with conservative systems; a comprehensive account of the stability and uniqueness results for the Cauchy problem for a systems of conservation laws can be found in 9 . In 3 ' 4 ' 5 and 6 a different problem was dealt with: let ue be a family of solutions to the parabolic systems u\ + A{ue)uex = eueXX' One expects that as e —> 0 + the solution ue converges in some sense to a solution of the corresponding hyperbolic system ut + A(u)ux = 0. The mathematical proof of this convergence was obtained via a suitable decomposition of the gradient of the solution ue along traveling waves. We refer to 6 for an account of the proof of the convergence of the vanishing viscosity approximation and of the uniqueness and the stability of the vanishing viscosity limit: it is important to underline, however, that the systems considered are not necessarily conservative, and no restrictions on the speed of the boundary are made. Here we consider the vanishing viscosity approach to prove the existence and stability of solution to (1). More precisely, we consider the system ' ut +A(t,u)ux
= euxx
u(0,x)
=UQ(X)
u(t,0)
= ub(t)
(3)
under the same assumptions of (1). We then prove the following theorem: Theorem 0.1. Consider the parabolic time dependent system ut+A(t,u)ux—uxx,
t,x>0,
(4)
with initial data UQ, ub with sufficiently small total variation and close to a constant u. Assume moreover that A(t,u) satisfies (2) for all u close to u. Then the solution u(t,x) to (4) exists for all t > 0 and has total variation uniformly bounded. Moreover, if u\{t), U2(t) are the solutions
15 of (4) with initial boundary data (ui,o,ui,b), (it2,0)W2,&) and with matrices Ai(t,u), A2(t,u), respectively, then fort > s IK(*) - W2(s)||ii(R+) < i ^ | t - S| + ||W1,0 - W2,o||Li(R+) + ||«i,6-U2,&||Li(o,.)+r°*-Var.(u) /" | | A I ( T ) - i 4 2 ( r ) | | L - d r ) (5) where L is constant depending only on the system (4) and the total variation ofAi(t), A2{t), u0, ub.
% (t/e,x/e). We prove that there is a unique limit to the solution constructed in Theorem (0.1), which is a viscosity solution to the hyperbolic system with boundary ut+A{t,u)ux=0.
(6)
Note that the system is not in conservation form, and the flux matrix A depends explicitly on time. As a particular case, we construct the solution to the boundary Riemann problem u0(x) = u0,
ub(t) — up.
(7)
This solution is a self similar solution, characterized by the fact that it is the limit of the vanishing viscosity solution. We prove the following theorem: Theorem 0.2. Consider the parabolic time dependent system ut + A(t,u)ux
=euxx,
t,x>0,
(8)
16
with initial data UQ, ub with sufficiently small total variation and close to a constant u. Assume moreover that A{t,u) satisfies (2) for all u close to u. Then the solution ue(t,x) to (4) converges as e —> 0 to a unique BV function u(t), the vanishing viscosity solution to (6) with initial data uo and boundary data ub. Moreover, ifu\{t), u2(t) are the solutions of (6) with initial boundary data (tti.ojWi^), (u2,o,U2,b) and with matrices A\{t,u), A2{t,u), respectively, then for t > s ||ui(t) -
W 2 (S)||L 1 (K+)
< L\\t-s\
+ ||wi,0 -
+ IK& - U2,b\\v(o,.) + TotVar.iu)
-«2,O||LI(R+)
J P i ( r ) - ^ 2 (T)||z«dr)(9)
where L is constant depending only on the system (4) and the total variation ofAi(t), A2(t), u0, ub. The case of two boundaries at x = 0 and x = L has been studied in 19 for a special 2 x 2 hyperbolic system. Since L can be arbitrary, the single boundary case follows by taking the limit L —» oo. The proof is base on the following arguments. First of all we study the ODE for boundary layers ux=
p
px = A(K,U)P KX
=
(10)
0
for u, K close to u, k respectively. This means that we have frozen time. The strict hyperbolicity implies that in general there are k — 1 eigenvalues of A(K,U) strictly less than 0, the fc-th eigenvalue is close to 0 and the other n — k are strictly greater than 0. Without any loss of generality we assume that \k{u = u, K = 0) = 0. By a careful decomposition, based on the exponential dichotomies on the system (10), we obtain that there is a matrix valued projector Rf, which describe the part of the solution to (10) which decreases exponentially fast. Equivalently we can extract from the solution the part of the boundary layer which decrease exponentially. The remaining part is described by the vector fk, which is the one dimensional part of the boundary layer decaying at +oo with an algebraic rate. Next we write the equations satisfied by the decomposition of ux in traveling profiles and boundary layer, Ux = Rb(K, U, Vb, Vk)vb + Vkfk(K, U, Vb, Vk,ak) + ] P Vif^K, U, Vi,0-i),
(11)
17
ut = Rb(K, u, vb, vk)wb+wkrk(K,
u, vb, vk,ak)+^2(wi-XifiVi)fi{K,
u, vi; at),
ijtk
(12) and the variable <j{ is given by /
°% = Kto-9
\X
x
(— J,
\x\<S0
0(x) = < smooth connection
50 < x < 350 (13)
9 is a cutoff function, and which implies the Lipschitz dependence of the solution u w.r.t. initial boundary data. When passing to the limit e —> 0 in (8), the main step is understanding the boundary Riemann problem (7). In fact, due to finite speed of propagation, the part inside the domain can be studied as in 3 , i.e. as a hyperbolic system in the full space. We construct the solution u(t) of the boundary Riemann problem, i.e. of the hyperbolic problem ut + A(K, U)UX = 0 u(0,x) =u0 u(t,0) = ub
(17)
which is the limit of the vanishing viscosity approximation: this means that the solution of this problem will be the unique L 1 limit of the solutions to
{
ut + A(K,U)UX u{0,x)
= euxx = u0
(18)
u(t,0) — ub as e —> 0. Observe that since the Boundary Riemann Solver is defined when the parameter K is constant, we can neglect the dependence of A from K. The solution to (17) is a BV self similar solution of the form u(t,x) = u(x/t),
x, t > 0.
Let u be the limit point of u as x —> 0: u=
lim u(t,x),
t > 0.
(19)
x—>0+
Since u is a BV function, this limit exists and is constant for t > 0. This point is determined in the following way:
19 (1) uj) is connected to u by a characteristic boundary layer, and waves (shocks or contact discontinuities) of the characteristic family with the same speed of the boundary, i.e. 0 in our case; (2) the Riemann problems [u, uo] is solved with waves of the families i > k which have a speed strictly grater than the speed of the boundary. Using the results on the uniformly stable manifold, we can write more explicitly the composition of the Boundary Riemann problem: (1) the uniformly exponentially stable boundary profile, which is given by the reduced ODE on the uniformly exponentially stable invariant manifold, coupled to a wave of the characteristic field entering the domain. These waves generate the boundary layer, and connect Ub to some point u\; (2) waves of the boundary characteristic family k with the same speed of the boundary, but not generating any boundary layer. These waves in the parabolic system do not travel with speed a^ = 0, because of the interaction with the boundary, but in the hyperbolic limit this interaction disappears being due to diffusion. We thus arrive to the point u; (3) waves of the boundary characteristic field k with speed strictly greater than the speed of the boundary, connecting u to some point (4) waves of the characteristic fields i > k entering the domain, connecting t*2 to uoThe part 1) generates the boundary layer, while the remaining parts can be obtained by means of the standard technique of 2 . The fundamental point is thus to construct the boundary layer for any small Boundary Riemann problem [ub,uo]. This construction is made by means of a contraction argument. We observe that since we are in the boundary characteristic case, the previous results on existence of boundary layers do not apply. References 1. D. Amadori. Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA, 4:1-42, 1997. 2. S. Bianchini. On the Riemann problem for non-conservative hyperbolic systems. Arch. Rat. Mech. Anal, 166(l):l-26, 2003.
20
Figure 3. The construction of the Boundary Riemann solver. 3. S. Bianchini and A. Bressan. BV estimates for a class of viscous hyperbolic systems. Indiana Univ. Math. J., 49:1673-1713, 2000. 4. S. Bianchini and A. Bressan. A case study in vanishing viscosity. Discrete Contin. Dynam. Systems, 7:449-476, 2001. 5. S. Bianchini and A. Bressan. A center manifold technique for tracing viscous waves. Comm. Pure Applied Anal., 1:161-190, 2002. 6. S. Bianchini and A. Bressan. Vanishing viscosity solutions of non linear hyperbolic systems. Ann. of Math., 161:223-342, 2005. 7. A. Bressan. Global solution to systems of conservation laws by wave-fronttracking. J. Math. Anal. Appl, 170:414-432, 1992. 8. A. Bressan. The unique limit of the Glimm scheme. Arch. Rational Mech. Anal, 130:205-230, 1995. 9. A. Bressan. Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford, 2000. 10. A. Bressan and R. M. Colombo. The semigroup generated by 2 x 2 conservation laws. Arch. Rational Mech. Anal., 133:1-75, 1995. 11. A. Bressan, G. Crasta, and B. Piccoli. Well-posedness of the Cauchy problem for n x n conservation laws. Mem. Amer. Math. Soc, 694, 2000. 12. A. Bressan and P. Goatin. Olenik type estimates and uniqueness for n x n conservation laws. J. Differential Equations, 156:26-49, 1999. 13. A. Bressan and P. LeFloch. Uniqueness of weak solutions to systems of conservation laws. Arch. Rational Mech. Anal., 140:301-317, 1997.
21 14. A. Bressan and M. Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete Contin. Dynam. Systems, 6:673-682, 2000. 15. A. Bressan, T. P. Liu, and T. Yang. L stability estimates for n x n conservation laws. Arch. Rational Mech. Anal., 149:1-22, 1999. 16. C. Donadello and A. Marson. Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws. Preprint, 2005. 17. J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 18:697-715, 1965. 18. J. Goodman. Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. PhD Thesis, California University, 1982. 19. L. Spinolo. Vanishing viscosity solutions of hyperbolic systems with two boundaries, to appear in Indiana Math. J.
A N OPERATORIAL D E S C R I P T I O N OF STOCK M A R K E T S *
F. BAGARELLO Dipartimento Universita
di Metodi e Modelli Matematici Facoltd di Ingegneria, di Palermo, I - 90128 Palermo, Italy E-mail:
[email protected] We review here some recent results concerning an operatorial approach to a stock market which is described and analyzed using the same framework adopted in the description of a gas of interacting bosons.
1. Introduction We review here some recent results concerning an operatorial, i.e. non commutative, approach to a stock market 1 , along the same line proposed by other authors 2 ' 3 . This point of view originates because of some simple remarks: the total number of shares in a market does not change in time, and the price of a single share does not change continuously, but for integer multiples of a certain minimal quantity, the monetary unit. Quantum mechanics (QM) provides a natural framework in which these features can be taken into account. In particular it also provides natural tools to discuss the existence of conserved quantities and to find the differential equations of motion which drive the portfolio of each single trader, as we will see. 2. The model The model we discuss in this section is based on the following assumptions: (1) our market consists of L traders exchanging a single kind of share; (2) the total number of shares, N, is fixed in time; (3) a trader can only interact with a single other trader, i.e. the traders feel only a two-body interaction; (4) the traders can only buy or sell one share in any single transaction; "This work has been financially supported in part by m.u.r.s.t., within the project Problemi matematici non lineari di propagazione e stabilita nei modelli del continuo, coordinated by Prof. T. Ruggeri.
22
23
(5) the price of the share changes with discrete steps, multiples of a given monetary unit; (6) when the tendency of the market to sell a share, i.e. the market supply, increases then the price of the share decreases. For our convenience, the supply is expressed in term of natural numbers and the monetary unit is normalized to 1. The formal hamiltonian of the model, which is assumed to describe the dynamical behavior of the market via the Heisenberg equations as in QM, is the following operator: H — H0 + Hi, where Ho = J2t=i aia\ai + Su=i Picl°i + °1 ° + P] P Hi =Y,lj=iPij [Aaj{cic])P + aia]{cj4)P) +{o]p+p]
(1) o),
where P = p^p is the price operator. Here the following commutation rules are assumed: [aual] = [ci,ci\ = 5inl,
\p,p*] = [o,ct] = I,
(2)
while all the other commutators are zero. We also assume that pu = 0, i.e. a trader cannot interact with himself. Here the operators a\, p", c\ and o" are respectively the number, price, cash and the supply operators. The states over the algebra of the model are w
{n};{fc};0;M( • ) = < f{n};{k};0;M,
• V{n};{k};0;M
>,
(3)
where {n} — n\,ri2,- • • ,ni, {k} =fci,&2,.. • ,k^ and
^ W ; M ; ° ; M •=
^ni\...nL\kl\...kLWfW.
^
W
Here <po is the vacuum of the model: a,j<po = Cjipo = pipo = otpo = 0, where j = l,2,...,L. For reasons which we have discussed in 1 , and which are related to the impossibility to give a rigorous meaning to operators such c?, it is convenient to replace H with an effective hamiltonian, H, defined as
H = HQ + HI, where Ho = D i i i aia\ai + £, L = 1 ftcjct + oU + p^p Hi = E f j = i P i i (4^(cic])
M
M
+ aia](cic\) )
(5) + (o^p + p^o),
24
where M — U{ny,{ky,0;M{P)- The following integrals of motion exist: L
L . N = ^^ a\ a,i, K — 22 clci
r = cto+p^p, and
1 Qj = a] % + — c] c,,
(6) for j = 1,2,... ,L. This can be easily checked since the canonical commutation relations in (2) imply that [H,N] = [H,t] = [H,K] = {H,Qj\=Q ioij = l,2,...,L. ^ The fact that N is conserved clearly means that no new share is introduced in the market. Of course, also the total amount of money must be a constant of motion since the cash is assumed to be used only to buy shares. Since also T commutes with H, moreover, if the mean value of cfio increases with time then the mean value of the price operator must decrease and vice-versa. This is exactly the mechanism assumed in point (6) at the beginning of this section. [H, Qj] = 0 finally implies that for each trader a given linear combination of the cash and of the number of shares stay constant. The hamiltonian (5) contains a contribution, hpo = cfi o + p^ p + (o^p + p^o), which is decoupled from the other terms. Solving the related Heisenberg equations we get o(t) = \{o{e~2lt +1) +p(e~2zt — 1)} and p(t) = \{p(e~2it + 1) + o(e~2it - 1)}. It is now trivial to check explicitly that both A(t) = o(t) -p(t) and f(t) = o t (i)o(t) +pHt)p(t) do not depend on time, so that A is another integral of motion. If we now compute the mean value of the price and supply operators on the state ^{n};{k};0;M w e get PS) = ±{Pr + Of + (Pr - Of) cos(2i)} Of(t) = \{Pr + Of- (Pr - Of) cos{2t)},
[
'
where we have called Of(t) = ^{n};{k);0;M{°\t)o(t)) and Pr(t) = w t {n};{fe};0;M(P (t)p(t)). Recall now that Pr — Pr(0) — M. Equations (7) show that, if Of = Pr then Of(t) = Pr(t) = Of for all t while, if Of ~ Pr then Of(t) and PS) a r e almost constant. In the following we will replace PS) with an integer value, the value M which appears in the hamiltonian (5), which is therefore fixed after the solution (7) is found. This value is obtained by taking a suitable mean of Pr(t) or working in one of the following assumptions: (i) Of = Pr; or (ii) Of ~ Pr or yet (iii) | 0 / + P r | > \Pr-Of\. In these last two situations we may replace Pr(t), with a temporal mean, < PS) >> since there is not much difference between these two quantities.
25 Let us now recall that the main aim of each trader is to improve the total value of his portfolio, which we define as follows: flj(t) =7nj(t)
+ kj(t).
(8)
Here we have introduced the value of the share 7 as decided by the market, which does not necessarily coincides with the amount of money which is payed to buy the share. As it is clear, II|(£) is the sum of the complete value of the shares, plus the cash, and it can also be written simply in terms of hj (t) as ftj(t) = I%(0) + (7 - MKhjit)
- n,(0)).
(9)
In order to get the time behavior of the portfolio, therefore, it is enough to obtain fij(t). The easiest way to find it is to develop the following simple perturbative expansion, well known in QM: n,-(t) = eiH%e-iHt
= hj + it[H,n,} + {^f[H, n , ] 2 + ^-[H,
n,]3 + ...,
(10) where [iJ,n,-]i = [H, hj] = Hhj — hjH and [H,fij]n+i = [H,[H,hj]n] for n > 1, and then to take its mean value on a state U{n};{k};0-,M u P to the desired order of accuracy. This perturbative analysis has been undertaken in 1 for a small number of traders and will not be repeated in this short review. Here we consider a situation in which the number of traders is very large, virtually divergent. For that it is convenient to take M = 1 in (5). This is not a major requirement since it simply corresponds to a renormalization of the price of the share. It is clear that all the same integrals of motion as before exist: N, K, T, A and Qj = hj + kj, j = 1, 2 , . . . , L. They all commute with H, which we now write as ' H = h + hpo, where < h = YA=I ^ni + Ef=i Pih + ££j=iPij [4ajci , hpo = o+ o + pt p + ( 0 t p + p t 0 ) )
4
+
ai a c
\i
c
i)
(n)
For hpo we can repeat the same argument as in the previous section and an explicit solution can be found which is completely independent of h. In particular we have W{n};{k};0;M{P) ~ 1- F° r this reason, from now on, we will identify H only with h and work only with this hamiltonian. Let us introduce the operators Xi = Oj cj, i — 1,2,..., L. The hamiltonian h can be rewritten as L
L
h = Y, («i«i + Pih) + Yl Pa (xi xi + xlXi) 1=1
i,j=l
•
( 12 )
26 The following commutation relations hold: [Xi, XJ] = 6ij(ki - hi),
[Xi, hj] = 6ij X»
[Xit kj] — -6ij X»,
(13)
which show how the operators {{Xj, X], hi, fcj}, i = 1,2,... ,L} are closed under commutation relations. This is quite important since it produces, after few minor computations, the following system of differential equations 1 :
al)Xl+2iXJL)(2hl-Ql) -*, 0 are all affected by inertial terms and we are interested in their effects. We introduce the internal moments, defined as (1) with the peculiar velocity C, = Ci — Vi instead of c*. vt is the velocity of the gas. Thus, p, Pij, and ^jp- are the mass density, the pressure tensor and the heat flux respectively, while pi is identically zero and pu is the density of internal energy. The F's can be expressed in terms of the p's by the relation Fiii2---iN
~ / j I i. I P(iii2...iN-kViN-k+i---viN)fe=0 ^ '
\"/
We choose a theory of 13 moments. In this case the first 13 equations of (2) in terms of the p's read b
T& + ^
p(*3t-i°i-Wikvk)=0,
- 4pk«Wj)k
+PiJtl+d-§^+
d
-T-ePk(ilwl)k+Pijj^ +3
P(H (iTT - tf) ~
2
PHi8^
= - r p
= 5±,
a = 4.8 1 0
9
^
(10)
and we solve the equations (8) for the boundary data ?M(rO = 1 0 3 $ ,
p(re)=s£,
v [0] (r e ) = 0,
v [0] (n) = 0.
&(re) = 300K±,
We choose He 4 as the gas, so that m in (11) is the atomic mass of helium. The prescription ( l l ) i of the heat flux at r» means that the parameter C in the equations equals 5 ^ . We choose x in the pressure relation (11)2 as 10, 20, 30, 50 and 100; all pressure values are thus appropriate to a rarefied gas. The set of equations is numerically solved and the solutions are shown in Fig. 2. Inspection of the figure shows that the azimuthal velocity lags behind the velocities of the cylinders in the bulk of the gas; more so the lower the pressure p (r e ) is. The velocity field has a narrow boundary layer near the outer cylinder. In order to appreciate the amount of lag we note that, with the data given, the outer cylinder has a speed of 0.1 ^ with respect to an inertial frame. The temperature increases toward the inner cylinder, as expected, since there is an outward heat flux imposed by the boundary conditions. At the inner cylinder the temperature has the value 306.9K, up 6.9K from 300K at the outer cylinder. For the lowest pressure p(re) = 1 0 ^ the temperature increase is a little smaller.
40 A
P(re)
p(re)
-0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035
a) v[&](r)
Figure 2. Solutions for values p(re) = (10,20,30,50,100) -^- increasing as indicated. a)v{d\{r) in f; b) g[tf](r) in ^ - ; c) T (r) in units of 300K; d) p(r) -p(re).
T h e azimuthal heat flux q [6] has a boundary layer t o match the boundary layer of the velocity field. Note t h a t , by (8)3, b o t h b o u n d a r y layers essentially compensate each other so as t o ensure the vanishing of the shear stress. For an appreciation of the values of q [9] ( r ) , shown in Fig. 2, we compare t h e m with q [r] (rj) which, by ( l l ) i , amounts t o 1 0 3 ^ . T h u s the azimuthal heat flux is mostly smaller t h a n 1 ° / o o of the radial one in the circumstances considered. T h e pressure grows with increasing radius because of t h e centrifugal forces. T h e heat flux makes the pressure profile concave rather t h a n convex, which it would be without heat conduction. References 1. I. Miiller, Arch. Rat Mech. Anal. 45, 241 (1972). 2. I. Miiller I. and T. Ruggeri, Rational Extended Thermodynamics (2 n d ed.). Springer Tracts in Natural Philosophy 37, New York (1998). 3. E. Barbera, I. Miiller, Inherent Frame Dependence of Thermodynamic Fields in a Gas. To appear on Acta Mechanica (2005).
N O N L I N E A R STABILITY OF U N I F O R M S T E A D Y STATES A N D PRESERVATION OF SIGN FOR A T W O - D I M E N S I O N A L F O U R T H - O R D E R REACTION DIFFUSION EQUATION
M.V. BARTUCCELLI Department University
of Mathematics
of Surrey,
E-mail:
Guildford
and
Statistics
GU2 7XH,
England
[email protected] In this paper we study the nonlinear stability of uniform steady states of a fourthorder reaction diffusion equation in two spatial dimensions. An important consequence of this stability is that the equation preserves the sign of a set of its solutions provided some appropriate restrictions on the parameters and the initial data are met. This equation and related ones are also used in the contexts of population dynamics, where obviously the solutions must be nonnegative functions. They have a very rich dynamics including travelling waves, periodic patterns and localised structures.
1. I n t r o d u c t i o n
Partial differential equations (PDEs) are one of the most useful and reliable way to describe a huge variety of real world phenomena. In particular they also often appear in the description of physical or biological contexts where their solutions evolve in such a way to preserve some fundamental properties such as positivity (think for example of evolution of densities). As it is well known, preservation of sign in solutions of dissipative PDEs of the second order can generally be achieved by using the maximum principle 1 . However this powerful method cannot be used in PDEs of higher order then the second. Therefore other techniques must be imployed for showing that in some circumstances some solutions of PDEs of higher order than the second do preserve their sign. In this paper we shall discuss nonlinear stability of uniform steady states and positivity of solutions for the partial differential equation in two spatial
41
42
dimensions on the unit square Q = {(x, y) G [0,1] x [0,1]}, ut = -aA2u2
Au + u(l -u2),
(1.1)
2
d d U ux where A = 7—1^7 + -7-^1 = ( : V, t)i t > 0, subject to the initial condition ox ay1 u(x, y, 0) = uo(x, y) and periodic boundary conditions. The parameter a is positive. Equation (1.1) is a generalisation of the well-known KolmogorovPetrovski-Piscounov-Fisher equation 2 which is used in population dynamics. It is closely related to the well known Swift-Hohenberg equation 2 . In the context of positivity of solutions and attractor dimension it has been already investigated in the case of one spatial dimension8. We shall use energy and ladder methods 3 ' 4 ' 5 to obtain estimates for various norms and their derivatives of the solutions of our PDE. Moreover, our strategy for studying the nonlinear stability of uniform steady states and hence positivity preservation, involves centring the equation on a non zero uniform steady state; here we choose the constant solution u(x,y,t) — 1 as we are interested in proving preservation of positivity (see below). We then analyse the solutions of the "centred" equation by finding estimates for their L°° norm so that to show the solutions of the transformed equation are bounded, in absolute value, by 1. More precisely we introduce v(x,y,t), defined by u(x,y,t)
= l + v{x,y,t)
where u satisfies (1.1). If we can show that v(x,y,t) H-,*)||oo< 1 su
for
(1.2) satisfies
alii
(1.3)
v x
where \\v(-,t)||oo = P(x,y)en \ ( >y>t)\ then u(-,t) is a nonnegative function for all t. If we can show that ||u(-,t)||oo —* 0 as t —> 00 then we have uniform convergence. 2. Convergence and Positivity of Solutions We begin our analysis by stating the classical notation for the seminorms
«i+n2=n
2
il
n1+n2=n
x
»
/
where v denotes the smooth function defined by (1.2). Substituting (1.2) into (1.1) we obtain for v the equation vt = - a A 2 v - Av-2v-3v2-v3.
(2.2)
43
Our aim is now to find conditions which will ensure that I M ^ —> 0 as t —* oo which establishes uniform convergence of solutions of (1.1) to the u = 1 solution thereof. We first obtain appropriate estimates for the L 2 -norm of the solution of (2.2) and also of its derivatives. Using these estimates together with a new interpolation inequality, we then estimate ||v|| in a way that only involves the parameter of our equation and interpolation constants whose values are explicitly known and sharp (which is very important). We start our analysis by investigating the evolution of the L 2 -norm of the solution v of (2.2), namely Jo- Differentiating Jo with respect to time and inserting the right hand side of (2.2) gives
- Jo = -a J 2 + Ji - 2 Jo
I / vsdxdyJn
[ v4dxdy
(2.3)
JQ
where the dot denotes differentiation with respect to time. To simplify the notation, from now on we drop the subscript Q in all the integrals but it is understood that all the integrations are done on the unit square Q,. The —01J2 and the J\ terms have been obtained by integration by parts (notice that the integrated terms vanish since the boundary conditions are periodic). By using Jj_ < J 2 1/2 J 0 1/2 < f J 2 + ^ J 0 and - 3 J V < ^WvW^fv2 = 3||f |oo-7o> equation (2.3) becomes
\jo
< -f ^ - (
2
~^)
J
o + 3 H U J o - JvA.
(2.4)
Next, the term ||v||c>o has to be estimated. To do this we shall use the following sharp interpolation inequality 6
Hoc < 4= MrtMv)*
+ Mv)*.
(2-5)
V71" Using these estimates in (2.4) yields
» .fe s _ » A - („ - i-)
Jo +
3-L ( o A ) 1 (|) " + 3 4 - Jl (J.6)
44
Next, by using Young's inequality we finally obtain: 3
2a/
4 \\/TTJ
2
V a
:= /(Jo) (2.7)
We need to study the solutions of the above differential inequality. By elementary theory they are bounded above by the solutions of the onedimensional ODE Jo = 2/(Jo). Note that the solutions of (2.2) certainly cannot all satisfy v —> 0 as t —» oo since this equation has uniform steady states v = — 1 and v = —2 in addition to » = 0. These other states correspond to the uniform steady states u — 0 and u = — 1 of (1.1). At these states one can check that / is positive. In fact, provided a > 1/4, the function / is negative for Jo small, and for Jo large, and positive in some intermediate range. We may therefore state that if Jo(t — 0) < J*, where J* is the smallest positive root of / ( J ) = 0, then Jo —> 0 as t —> oo. Sufficient conditions are that Jo(t = 0) < 1 and
-i4(^(!)W -4-
™
In order to have control on the L°° norm of the solution we need to have estimates on the behaviour of J\ and, more importantly, on J2 as well, because we are in two spatial dimensions. We start with the analysis of J1. Its evolution equation is \j\
= - " ^ 3 + J2 - 2Ji - 6 /' v{Dvf
- 3 fv2{Dv)2
(2.9)
and, after some transformations similar to the ones used in obtaining the estimate for J 0 , and by using the Cauchy-Schwarz inequality and Young's inequality we arrive at 1 .
a f 1 - 7 ^ 3 + 1 1 + T - J\. 2 2 V 2a 3 JBy using the inequality 7 ' 5 — J 3 < — —^ we finally obtain 0
J
l
^ oo don't give independent equations, because from eq.(4) it follows that lim^oo Fl^-tr = limc >00 Flf^-'tr. In other words, pn—*r _ F^---1*- i s higher order infinitesimal with respect to c _ 1 , so that we have to find a suitable linear combination of eqs. (5) and multiply the result by an appropriate power of c, before taking the limit. This will be done in the next section. 3. A new form for the system (5). Let us consider the numbers bhr= (-l)h (™\ ( n +
w
- | l ) ! 7 ? (N - M -2n,N -M -2n + 2h- 2)(8)
[n + my.
\ nJ
where 77(12, b) denotes the product of all odd numbers between a and b if 1 r JV-l-rl „_. _ \M-\-r~] a < b, while it is 1 if a > b; moreover n = [—~ m 2 ~ ], Obviously, bor = 1. In the appendix the following will be proved Proposition 3.1: The numbers defined by eq. (8) satisfy the equations m ^2 hrCn+j-h = 5j,m+lbr, for j = 1, . . . , TTl + 1 with (9) h=0 + 2,N-M) , (10) Ch = ±-r](N-M-2h 6r = ( - l ) m
"'
.,,, ,'
.v(N-M~2n,N-M
+ 2m). (11)
Let us also consider the numbers Ofer = -
Yl h=0
b
hrCk-h, for fc = 0, . • . , [N~2~T]
•
(12)
50
After that, let us consider the following linear combination of F^'"ira of F}}-i'a: ~i1---iraeiei---ejV4-M-i-2r ew+M-i-ir
and
v 2r M" ireiei "" e ' e *°(- 2c2 )"
£ 9=0
a
prrN
\
Z C
(-2c 2^)
/
M-t-JV-1-272
(13)
p=0
where the index a has to be omitted if it is zero. Note that this tensor has N + M — 1 — r > N — 1 indices (if a = 0) so that there is no possibility of confusing it with Fl^'"%r. The corresponding linear combination of eqs.(5) gives eqs.(2)2, while eq.(2)i is eq.(5)i except that now the index N has been omitted. Obviously, we also define -1
=
Q
r M —1—ri L 2 J
r6
E
r
—-?M
+
9=0
2
1 6r
a
V(-2c)
^
(-Id1
N+
M-l-2r-
ji1---ireie1--epep
(14)
p=0
where the property [ £ = £ = ] + [ M ^ = r ] = JV+M 2 ~ 3 ~ 2r has been used (it is a consequence of the fact that N + M is odd). The interesting thing, which we now prove, is that limF
\---iTae\e\---e N+M-i-2r e N+M
C—>OQ
,
-
T
, . - , . . ,
—=//«il • • • ^ >
z i - - ^ r a e i e i - - - e ^+M~l-2r&N
2
+
)^^%,(15) M-l-2r
and we indicate this limit by r 2 2 ; moreover, ua is 1 if a = 0, is uk if a =fcand / = limc^oo m^f, as in [2] (in the sequel the factor m^ does not affect the results, so we will omit it). To prove eq.(15), we see that (13), by means of (4) gives ~H---irae1ei---e N+M-I-2T
t
2
7M-W
J2 9=0
e N+M-I~2T
2
bqr{u2)q{-2c2)-q
1
=
(-2c 2 )^^//V + 4 u
+ J2 apr{u2f{-2c2)-P
* •••u 'u du.
(16)
p-0
Let us denote the expression between square brackets as [•••]; by inserting N-M
the expansion of ( 7 ) ^ = ( l - £ ) '
= £^=0
^ch(u2)h(-2c2)-h
51
it becomes [•••]= [co + c 1 u 2 ( - 2 c 2 ) - 1 + . . . + ch(u2)h(-2c2)-h]
• rJV-il 2
r M —1
,2
bor + birU ( —2c )
oo
+ •
h brAf-i-ri
inf{fc,[^^]} " "v*>L2 J/
•(-*)-" = £
/,
,2
N \
2 Ofcr(«.2\* )*
+ ^
-2c2
fc=0 fc
K
l
2
'
/
2
\
k
E ^ - f e ) + E ^(-^) •
fe=0 h=0 \ ZC J \ ZC J k=0 iv 1 r Now, the tensor for k < [ ~2 ~ ] disappears for eq.(12), while those with [^Y^] + lv> t n a t a r e solutions of the (rescaled version of) MeF 13 system . ^
=
_«*
+ ( K + y )
^ _ ^
K
^ ( 1 )
2
k
jAi>kv + (Vv + V)i>kv -e2K
i e ^ =
^
where K ~ P • V V, P is the interband momentum matrix, V is the electrostatic potential, Vc, Vv are the minimum and maximum of the conduction and the valence band energy, respectively, and e is the Planck constant. Then, by using the Madelung ansatz ipk = ynb e x P i^b/e) w ^ n the bandindex b = c,v, the hydrodynamic system corresponding to Eqs. (1) reads -^r + divJk = -2eK Im n\cvi dnk dJk
dWJk =
2eK
,. (3kc®Jk\ =
Imntcvi 2
k„fe
AJ^c\
kmr
e'Vif Re nk„ + (K Re « ( u j - a ' ) )
e2VK Re nkcv - eK Re (nkcv(ukv - uk))
eW
jk °v
nk,
jk _ ^c_
n£
(2)
55 where Jbfc = n£V££, ukb = e V y ' n g / ^ + % j£/n%, ak = {Sk - Sk)/e and n cv ~ V^cV^v exp(ic7fc). In the mixed-state description 8 , densities and currents corresponding to the bands are rib •'= Ylk^knk,Jb '•= J2k^kJb' Ub := eVi/nb/y/nb+z Jb/rib = euos,b+iue\tb, while the "interband" quantities are a := Ylk^kCF anc ) - n c V( J \
„ ;__ 2^/nc
)+ ncW
= e2VK Re Rcv +e2KRe at
- div
\
nv
h n-A
J
+ n„V
\
1_ 2v/n^
= e2VKReRcv
with # c „ = ^2k^knkv
, Q c = Yjk^knkv
Qcv,
(3)
+ n„W - e2K Re Qc
[uk ~uk).
In analogy with the
8
one-band case , we introduce the temperatures 6b = d0s,b + #ei,6) 6 = c, w, with the osmotic parts #OSjb defined by Qos,b = >
,ik Afc — ( u „ s b - Uos.b) ® («os,6
_
u
os,fc)
and the current temperatures #ei,& defined correspondingly. If we call a:==]TA^,
^ : = ^ A f c ^ ( 4 - ^ ) , 0C := £ A fc ^(^ - uc),
the coupling terms contain Rcv — ancv , Q cv — Tlcv
[a(uv - uc) + 0V - p\\ . In order to find a relation between a, 0V and /3C and the hydrodynamic quantities, we take the gradient of a and use the definition of ncv, uc,uv and the identity 6 V Tlcv . / __ Jv Jc — uv — uc = % ( ever 1 fil!
fir
56
Accordingly e V ( 7
- ^ +^ = - ( e V a - A , - & ) .
(4)
The last equation of system (3) can be rephrased as
*+x>G!K)-i>(;H) (5) and, by comparison of (4) with (5), we get
^(|-£)-X>(|-£)->«-*-*). then Re {(eVa - (3V - /3C) /a] = 0. Accordingly, Eqs. (3) can be written as 071
~
+ divJc = -2eK Im
(an^),
—— — divJw = 2eK Im(«n c „), —^ + div + nc9c - n c V Y_ + n c W = e2VK Re (ancv) + e2K Re (ncv [a(uv -uc) + 0V-JQ),
(6)
- r - - div + nv9v + nvV 1_ + nvW at \ nv I \ 2Jny J = e2VK Re (ancv) - e2K Re (ncv [a(uv - uc) + Bv - &]), e V <x-^ + ^ = - I m j i ( c V a - A , - ^ ) } . *. nv nc [a J The terms on the right hand side of the equations determine the coupling. The system is not closed: the quantities a, f3c and f3v are not expressed in terms of the hydrodynamic quantites, but they are linked by
Re{(eVa-A,-&)/a}=0.
(7)
In addition, we must assign constitutive relations for the tensors 9C and 9V. A simple class of closure conditions can be obtained by assuming a = a(nc,nv,a) and taking „ da a Bc = 2nc-—uos,c dnc
da - "H-Uei.c, da
da Pv = 2nv-—uos dnv
da „ + —u e i,„. aa
(8)
Then eVa - (3V - Bc = 0, thus Eq. (7) is fulfilled and moreover eVcr Jv/nv + Jc/nc = 0. For the temperatures we assume ni,6b = Pb{rib)I, where I is the identity tensor and the functions pc,pv are pressures. In particular,
57
we can take pt, = 6°rib, a s f ° r a n ideal gas in isothermal conditions, and a = 1,/3C = (3V = 0 (which follows, e.g., from n^v ~ ncv). Accordingly, we obtain the simplest two-band isothermal QHD model - ^ + divJ c — -2eK at
Imncv,
—~ - divJv = 2eK Im ncv , at dJc JC®JC , ^ 0vT ^fe2Ax/7Tc -z- + div h #"Vn c - n c V I_ dt nc \ 2y/nc
F
\ )
= e2VK Re ncv + e2K Re (roc„ (uv — u£)), ,. Jv®Jy div
dt
7l v
(9)
, a0v7 , ^7(e2A^nv~ 1- 9 SInv + n B V | „ ;__ + K y
LyJTiv
e 2 VK Re ncw - e2K Re (nOT (u„ — u~c)), a
1
= 0.
Here, the only peculiarity of the mixed-state case is the presence of the temperature terms. 3. The drift-diffusion model Now we perform the zero-relaxation time limit of the hydrodynamic models recovered. First, we start from Eqs. (9), we add relaxation terms for the currents and we introduce the diffusive scaling t t > , jc TJC, JV —> TJV, (10) T
that leads to the ansatz a —> GQ + ra, where OQ is a constant phase to be determined. In the limit r —> 0 the system reads dnc + div J c = -2eK nc^/nva, dt dnv 2eK^/n^^/n^a, — divJy
~~d7
J, = - e°Vnc + nc l V
'e 2 A v / n^ 2y/n~c
K
r 2
Jv =
6°Vnv - nv I V
£7
1 Tly
=0, Tlr
e Av/n^
+ e2
-V V
K
(11)
58 where crn = 0, due t o t h e limit of the first equation. Alternatively, we start from the isothermal version of the Eqs. (6), closed with a = a(nc,nv), f3c := 2 n c J ^ L u o s ? c , 0V :— 2nvJ^LuOSiV, we add relaxation t e r m s for t h e currents and we consider the diffusive scaling in Eq. (10), with {x, t)) , *Work partially supported by a Focused Research Grant from the NSF #DMS-0354488 ••"Work partially supported by the INDAM and by the PRIN grant: "Nonlinear mathematical problems of wave propagation and stability in models of continuous media".
59
60
where y = 4>(x,t) is the curve at time t in the plane (x,y), and 7(1) is the tangential jump of the velocity at the point (x, (x)), see Ref. 9, 10. The dynamics of the curve on the plane is ruled by the celebrated Birkhoff-Rott equation:
|,-(r,,) = - i evf—JSL^
(1)
where the complex number z represents the complex coordinates on the plane, z* is the complex conjugate, and where the generic point P on the curve has been characterized using (instead of the coordinate x) the total circulation T between a reference point and P itself. Moreover the shape of the curve (j>(x, t) and the strength of the tangential jump are ruled by the equations 9 : dt(f> = -uidx(f) + u2 da = ~ldxui
- ui9x7 ,
(2) (3)
where Ui and u^ are the two cartesian component of the velocity of the sheet determined by (1). The well posedness of the Birkhoff-Rott equation is an interesting mathematical problem. One can show (see e.g. Moore 8 and Caflisch and Orellana 4 ) that in general a mechanism similar to the Kelvin-Helmholtz instability leads to the ill posedness of the Birkhoff-Rott equation. Instead, if the data are analytic, one can prove the short time well posedness of the problem 12 . Long time existence was proved for small analytic perturbations 3 , and taking into account the regularizing effect of surface tension 5 . All the above theory was derived neglecting the role of the viscosity. When this effect is taken into account one has that the diffusion of the vorticity would lead to consider vortex layers instead of vortex sheets. To model this effect Moore 7 studied the case of a vortex layer of small thickness of constant strength. The first order effects of the thickness on the motion of the layer were derived. Benedetto and Pulvirenti 1 proved rigorously that the dynamics of a vortex layer of constant strength converges to the dynamics of a vortex sheet when the thickness of the layer goes to zero. In this paper we shall consider the case of a thin vortex layer of non uniform vorticity in the limit of small viscosity. We shall suppose the layer to be of size the square root of the viscosity and derive the equations for the fluid motion inside the vortex layer. We shall prove that these equations are well posed for a short time when the initial data are analytic.
61 2. The Navier-Stokes equations in a moving curvilinear reference frame Consider the Navier-Stokes equations in the 2D domain A written in an inertial cartesian frame (x,y): dtU + U-VU
+ Vp = vAU
[0, 2TT] x
(4) (5)
where U = UiX + ViV is the fluid velocity, p is the pressure, V {dx,dy) is the gradient operator, and A = d% + dy is the Laplacian. In the domain A we now consider a smooth curve y = <j>{x, t). Following Moore 7 , we introduce the intrinsic reference frame (s, n), where s and n are the unit tangent and normal vector to the curve. We denote with s the curvilinear coordinate on the curve, and with R(s,t) the position of a point on the curve, see Fig. 1. Therefore the position of a point on the plane (close to the curve) is r = R + nn. The following formulas hold: d's ds
n p
dn ds
s p
where p is the radius of curvature of the curve. Moreover, denoting with djdr the time derivative in the comoving frame, one has: 5s dn fin — = -Sis dr or where Q(s,t) is the angular velocity of the frame (s, n ) .
Figure 1.
(6)
The comoving frame adapted to the curve.
The velocity of a point whose coordinates (s,n) are kept fixed is dTR + ndTfi; therefore the fluid velocity U can be written as 7 : U = us + vfi + dTR + ndTn .
62
If one introduces the decomposition: R = Xs + Yn , and uses the notation X = dTX, Y — dTY and (6), one has: X -Q,(Y + n)\s + Y +
dTr = dTR + ndTn=
flX\n.
= V + ft x r ,
(7)
where we have defined V = X's + Yn and ft = (0,0, £2). Therefore: U
u + X-
Cl(Y + n)
s + v + y + ax n
u + V + fl x r ,
(8)
where u = us" + wn is the velocity as measured in the comoving frame. Inserting the above equations in (4)-(5) and using the expressions for the differential operator in curvilinear coordinates reported in the appendix, one can write the Navier-Stokes equations in the frame adapted to the moving curve: dTu + Xu
n2X - (l(Y + n) +
2£l{v + Y)-
dsu + dsX - dsQ(Y + n)--(v
+ Y) + vanu + -— hi
v(A(u
dsv + dsY + dsnX + - (u
rixr))s
(9)
Cl2(Y + n) + ClX +
dTv + Y + 2Q,(u + X)u
+V +
+1) + vdnv + dnp =
hi v{A(u + V + nxr))n
(10)
^-{dsu m
(11)
+ dn[hiv}}=0
3. The vortex layer equations We now suppose that across a curve y = <j>(x,t), expressed in curvilinear coordinates (S,TI,T) as R(s,t) = X(s,t)s + Y(s,t)fi the fluid experiences a rapid variation. This variation occurs through a distance that is of the order the square root of the viscosity e = y/v. This means that, introducing the rescaled normal variable N = n/e, we shall suppose that 8NU — 0(1). Using the incompressibility condition one can see that the normal velocity has to be 0(e). We therefore rescale the normal velocity as v —> ev', with ONV' = 0(1). To keep the notation simple we rename v' as v. Therefore,
63
the equations that rule the flow (u,ev) inside the vortex layer read: dTu + X - 2QY - n2X -Q.Y + dsu + dsX -
Ydsn
p
vdN u + dspL =
(12)
ONNU
dNpL=0
(13)
dsu + 8N v — 0
(14)
u(s,N —» ±oo,t) -
(15) **(«,*) The second equation says that the pressure is constant across the vortex layer and can be recovered by matching with the value of the pressure of the flow outside the layer. The fact that the pressure is constant inside the layer is consistent with the continuity of the pressure across the layer. The fact that the pressure is continuous across the layer can be recovered through the same argument used to derive the continuity of the pressure across a vortex sheet (see Ref. 10 pp. 28-29). The matching value of the pressure can be calculated from the Euler equations. Using the Euler equations in curvilinear coordinates (i.e. (9)(10) with v — 0) calculated at the vortex layer n — 0 (from above and from below), one gets: -dspL=
dTu++ X- 2QY- £l2X - &Y+ dsu+ + dsX - ds£lY - Yp- i (16)
L
2
-dsp = dTu~+ X- 2QY- Q X - QY+ dsu+ + dsX - ds£lY - Yp~ (17) +
where with u and u~ we have indicated the matching values of the vortex layer tangential velocity with the outside Euler velocity that are related to U+ and U~ by the relations (see (8)): U+ = u+ + X - QY ,
U~ = u~ +X-QY
.
Equation (14) says that the normal velocity v can be recovered from the tangential velocity u through an integration: v = - /I dsu(s,N',t)dN' (18) dsu(s,N',t)dN' . Jo To equation (12) we impose an initial datum uo that must be compatible with the matching condition: u(s, N, t = 0) = u0(s,N)
where
u0(s, N ^ ±oo)—>u±(s,t
= 0), (19)
64
To have a closed equation it remains to be specified the motion of the curve as expressed by R(s,t) = X(s,t)s + Y(s,t)n. The curve is in fact convected by the vorticity field inside the layer which, to the leading order, is dnu. If one denotes with xs and ys the coordinates of the curve in the cartesian reference frame and introduces the complex variable z = xs + iys, one can see that, to the leading order, the dynamics of z (formally) is ruled by the Birkhoff-Rott equation (l)-(3), where 7 is the jump in the tangential coordinate U+ — U~. 4. Well posedness In this section we shall prove that eq.(12), with v expressed by (18), dspL given by (16) and (17), and with the initial condition (19), is well posed in an analytic function space. We shall prove the following Theorem which is the main result of this paper: Theorem 4.1. Suppose we have an analytic curve y = 4>o(x) across which the velocity field has a rapid tangential variation 7 = U+ — U~ with U+ and U~ analytic. Denote with u the velocity field as observed in the reference frame comoving with the curve having introduced the decomposition (8). Let u+ andu~ the values ofu corresponding to U+ and U~ in the decomposition (8). We can therefore suppose thatu^ G H/£,PO,T0- Suppose thatuo = u(t = 0) is such thatu0-u+(s,t = 0) G # (3Q, 5 < So, n < Ho and T < TQ such that there exists a unique solution u of the vortex layer equations (12)-(15), with u-u+ 0 Definition 4.2. The function space H£* + is the space of f(s,N) s G]sL,Sfi[ and AT £ [0, oo[, which are analytic w.r.t. s in D(5,SL,SR), w.r.t. iV and exponentially decaying for N —> oo, and such that:
with C2
2
1/1^,+= £
J2
SU
sup | | e " J v ^ ^ 7 ( . + i 9 S , i V ) | | L 2 ( R i ) < o o
P
The space function H™ _ is defined analogously and contains the functions f(s, N) defined for N £] — oo,0] and exponentially decaying for N —* —oo. Definition 4.3. The function space H™ „ T + is the space of f(s, N, t) with s £}sL(t), SR(t)[, N G [0, oof and t G [0,T], which are analytic w.r.t. s in D(5,si(t),SR(t)), C2 w.r.t. N and exponentially decaying for N —> oo, and C 1 w.r.t. £, and such that: |/k/i,j3,T,+ = 2
T
V
sup
sup
sup | | e ^ - W J v ^ ^ / ( - + ^ s , i V ) | | L 2 ( 3 i s ) +
^ < ™ _ ; 0 < t < T 0<JVS|0, Mj + acting on functions g(s,t), M^ and M^ acting on functions u(s, N, t) with N > 0, such that they solve the problems: {dt ~dNN
+ aNdN
)M0+u0 = 0,
N>0
(24)
= 0) =u0
N>0
(25)
M+u0(s,N,t M£u0(s,N (dt-dNN
= 0,t) =0
+ aNdN)M+g M+g(s,N,t M+g(s,N
= =
(26)
= 0,
N >0
(27)
0)=0,
A^>0
(28)
0,t)=g.
(29)
67
(dt -dNN
+ aNdN
)M+f = f,
N>0
(30)
M+f(s,N,t
= 0) = 0,
N>0
(31)
M?u0(s,N
= 0,t) = 0
(32)
while the operator M£ has the property that M^f analogously define the operators M~. 4.4. The vortex layer equations
= M^d^
in the operator
f. One can
form
Define the following operators: Ki[f]=2fdsf,
K2{f} = f[
dJdN'
(33)
Jo
K+[f] = u+dsf
+ fdsu+
+ f [dsX - YdsQ - Yp-1]
(34)
and G+[u+, u-,t] = M+Kilu+j + M+K2[u+] + M}K+{U+] +
+
M0 (u0 -u (t
+
= 0))+ M? [u~ (s, N = 0, t) + u~ - u+]
where now in M+ we have identified the function a with dsu+. One can define analogously the operator G~[u~,u+,t]. Using the properties of the operators Mf- one can see that equation (20) with the initial and matching data and with the boundary condition (21) can be put in the form u+ = — G+[u+,u~,t], with a similar form for the equation satisfied by u~. Therefore, defining u = {u~,u+) and the operator G[u,t] = — (G~[u~,u+,t], G+[u+,u~,i\) the two coupled equations for u+ and u~ can be put in the abstract form: u = G[u,t]
(35)
We solve this equation in the space H™^p , ± for each 0 < t < T and the following estimate holds:
\M3 h\m,^,± 0, /'(0) < 0, (3 being a positive constant such that ph is sufficiently small. We can write in Q: po(y) = /(0) + (3(y + b)f'(0) + ... . Then the liquid is called almost homogeneous in Q. We restrict ourselves to this case. In the sequel, we replace po(y) by the constant p and p'0(y) by the negative constant —Pp. We obtain, instead of Eq.(3) and Eq.(4), the approximative equations: U=--Vp-pg0xy-pgUvy
+ 0(yx-xy)
,
(6)
which is analogous to the Boussinesq equation of the theory of the connective fluid motion, and (Jo + Ji0)6 + p (' {xUv-yUx)dCl Jo. — (m 0 a + mi b)g 6 — g P g
=
x Uy dfl .
(7)
4. Operatorial equations of the problem and properties of the operators 1) We can suppose that U e J0(Q) = {UG C2(n) = [£ 2 (ft)] 2 , divU = 0, U-n = 0 on a} , U • n makes sense as element of // _ 1 / 2 (o r ), p £ H1^), and then Vp e G(ty, space of potential fields. Now, we consider the orthogonal decomposition of Weyl [2]: Let Po be the orthogonal projector from £ 2 (fi) into j7o(fi). Projecting Eq.(6) on j7o(^), w e eliminate p and we obtain U + KU + P g 0Po(x y)9 + P0{yx - xy)8 = 0 ,
(8)
74
where KU = f3gPo(Uyy); on the other hand, we can write Eq.(7) in the form Je + L& + P9MU+moa
+ mb99
= 0,
(9)
Jlo
where J = 1 + ^- , LU = ~ [\xUy~y Ux) dO , MU = ~- j xUyd£t. Jio Ji0 Jn Jio Jn 2) We must study the properties of the operators which have been introduced. a) The operator K from JQ{£1) into JQ(Q) has been studied in [1]. It is bounded, self-adjoint, definite positive and its spectrum (3g. In what follows, we suppose that the condition Eq.(12) is verified. 5. The point spectrum of the problem Setting U(x,y,t) = eiwtU{x,y), obtain the equations
9{t) = eiut9, from Eq.(8) and Eq.(9) we
KU + (3gP0(xy) • moo + mib g-Juj2>\e
u2P0(xy-yx)
(13)
= LO*U
= J2LU-(3gMU
.
(14)
First, we seek the eigenvalues w2 such that fig < LO2 < \\D\\. Setting \i = J1 we can rewrite Eq.(13): (/ - nK)U = [n(3g P0(xy) - P0(xy-
yx)
Since ||/iif|| < 1, I — ^K has an inverse (/ — fiK)~l = 7Z(/J,) holomorphic for |/x| < (fig)~l. Eliminating U between Eq.(13) and Eq.(14), we have Q(n) 9 = 0 , where Q(AO =
M
m0a + mib T
T
J Ic+^L+ppM}n(fi)[iiPgPo(xy)-Po{xy-yx)
The operatorial function Q(fi) is holomorphic for |/x| < {(3g)~x, selfadjoint. Q(0) is bounded from C into C, and therefore compact and we have
so that, by virtue of the condition (12), Q'(0) is strongly positive. Then, for the operator pencil Q(fi) in every interval [0,77], 0 < rj < (f3g)~x, using a general theorem in [2], there exists one and only one eigenvalue. Obviously, it is equal to ||D||.
6. T h e essential s p e c t r u m Now, let us seek the eigenvalues OJ2 such that 0 < LJ2 < (3g. The coefficient of 9 in Eq.(14) is strictly positive. Eliminating 9, we have [M(UJ2)-U2I}U
=0 ,
UGJO(Q.)
76
where
M{LO2)
= K - V{u>2) and
f"°0 + "'» g_jJT\Ic.
V(UJ2)
= (3gP0{xy) - uj2P0(xy-
yx)
(pgMU-^Lu).
Therefore, the problem is reduced to the study of the operator pencil M(UJ2)-UJ21.
For co2 < fig, the operator V(UJ2) is obviously self-adjoint and compact from So(Q) into v7o(^), so that we have classically: ae[M(uj2)} = and || • || the integral and the L 2 (fi)-norm. Finally we assume that < u >=< v > = 0, in order to guarantee the ms uniqueness.
79 3. Onset of penetrative convection localization In order to localize the zone in which the penetrative convection can occur we set ps(z) — p(Ts(z)). By virtue of constitutive law for the density, one obtains that p's{z) = -2aT'3{z) [T,{z)-4], p's\z) = -2aT's'{z) [Ts(z)-4] 2a [Ts(z)]2. We distinguish the following two cases i) Fluid layer heated from below. Since r = T\j — TL = 4 - TL < 0, it follows that 2/F + T > 0 =>• 3!z* € (0,1) : Ts{z*) = maxT s (z) > 4° C. Therefore the motionless density attains its maximum on z = 1 where Ts(z) = 4 and attains it minimum on the internal plane z = z*. The layer is divided into two parts: a) M2 x [0, z*\ "potentially stable"; b) M2 x [z*, 1] "potentially unstable". ii) Fluid layer heated from above. Since r = T\j — TL = 4 — Tj, > 0, it follows that: if
/?* - r > 0 => 3! z** G (0,1) : Ts(z**) = maxT 5 (z) > 4° C.
Hence the motionless density attains its minimum on the plane z — z**, z** £ (0,1) and its maxima on z = 1 and z = z, z e (0,z**) where Ts{z) = 4. The layer is divided into three parts: a) M2 x [0, z] "potentially unstable"; b) E2 x [z,z**\ "potentially stable"; c) R2 x [z**, 1] "potentially unstable". In both the cases, the fluid particles tend to redistribute themselves into the layer M2 x [0,1] in order to overcome the weakness in their arrangement: they tend to penetrate from the unstable layer(s) into the stable one (Penetrative Convection Phenomenon). To be sure that penetrative convection can occur, we assume —2(3* < r < j3*, and look for conditions guaranteeing the ms instability 13 ' 14 . On following the methodology introduced by Rionero 8 - 1 2 , we set £„ = n2ir2 + a2 and am = -Pr£n
,
a2n = 2RHPr°-[G(z)
- 1],
sn
' bln = -RHG'(z), . Ci„ = 1 ,
b^^GH^R,
(5)
C 3 „ = -T)£n ,
m
5^ = & , i=l
b2n = -£n,
m
s = jyt, i=l
m
um = ^ U l . i=l
(6)
80
By taking the third component of double curl of (4)i, with a procedure completely analogous to the which one of Section 5 of [10], one obtains =
E&~|r
Yl&(aiiWi + a 2 ^ ) + (V x V x (Um • VUm))) • k+
+(VxVx([sis)fk)-k ~
= f > i ^ i + hA + h^)
- U m • VS£~>
(7)
and the associated system dwi —— = auwi dt —-
89\ , + a2i0i, —- = bnwi + &2i0i + 621I1 at
= C n ^ i + C31I1
7j7
= Ol(m-l)^l +
fl2(m-l)Pm-l
— oT— = ^1(171-1)^771-1 + &2(m-l)^m-l +
dTm-1 TjjT
— Cl(m-l)Wm-l
,
r
+ C3(m_i)l
m
h(m-l)Fm-l
_i
dw-n 9i
£77.
+ — ( V x V x (U m • W m ) • k + ~(V
x V x ([5^]2k) • k
as
-^-=blmwm l dTdt
+ b2m9m + hmTm-Vm
•
VS^
•=cimwm + c 3 m r m - u m • vs^p
(8) Let ( u 1 , . . . , u m ; 6 ' i , . . . , ^ m ; r 1 , . . . , r m ) be a solution of (8), then (ui + . . . + u m ; 9i + ...+0m; r i + . . + r m ) is a solution of (7). Therefore - by virtue of the
81 uniqueness theorem - we can evaluate the behavior of (UJ, 9i, Tj) i = 1,.., m m
by substituting (8) to (7). In view of ||U m || 2 = ^ H U ^ 2 , \\S^\\2
=
t=i
X^II^H 2 ' H^m^l2 = 5Zll r i H 2 '
[t f o l l o w s t h a t t h e
instability is guaranteed
i=l
%=i
by the instability of the zero solution of —— —anwi d9i dt
+a2i0i (9)
bnWi + &21#1 + ^ 3 ^ 1
c
• -57- =
n w i +C3iri.
*• at The (9) eigenvalues equation is (10)
A3 + aiA 2 + a2(2;)A + a3(z) = 0 where ' ax = (Pr + r) + l){it2 a2(z)
+a2)
= (Pr + nPr + n)(TT2 + a 22\2 )2 -
a3{z) = T?Pr(7r2 + a 2 ) 3 - 2Pra 2
2Pr
67 7r2 + a 2
7r2 + a -
;[l-G(z)}G'(z)R2 [G{z)-l]R2 (11)
Being a2(z) = AiA2 + A!A3 + A2A3 if 8 3 ZQ € [0,1] : 0:2(^0) < 0, then exists at least one real positive eigenvalue or complex eigenvalues with real positive part. Theorem 3.1. Let 7 G (0,1) and R2>R2
(1 c
+ V + VP^LII 1 + 27
RE
(12)
hold, where RB = 27/4 7r4 is the classical Rayleigh number for the onset of natural convection. Then the penetrative convection occurs. Proof. By straightforward calculations and (12), one easily obtains that 3 z 0 £ [0,1] :a 2 (z 0 ) < 0.
82 R e m a r k 3 . 1 . We observe t h a t one has t o compare the critical number (12) with t h e which one coming from 3 z 0 e [0,l]:a3(zo) = A1+A2 + A 3 < 0 . We will consider this problem in a forthcoming paper.
Acknowledgments This work has been performed under the auspices of the G. N. F . M. of I.N.D.A.M. and M.I.U.R. (COFIN2005): "Propagazione non lineare e stability nei processi termodinamici del continuo". Prof. B. Straughan, who drew our attention on the Krishnamurti problem, is warmly acknowledged.
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11.
12. 13. 14.
M.A. Azouni, Geophys. Astrophys. Fluid Dyn. 24, 173 (1983). M.A. Azouni, C. Normand, Geophys. Astrophys. Fluid Dyn. 24, 209 (1983). M.A. Azouni, C. Normand , Geophys. Astrophys. Fluid Dyn. 24, 223 (1983). J.N. Flavin, S. Rionero (1996) Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton, Florida. M. Gentile, S. Rionero, Rend. Ace. Sc. fis. mat. Napoli LXVII, 129 (2000) . R. Krishnamurti, Dynamics of Atmosphere and Oceans 27, 367 (1997) . L.E. Payne, B. Straughan, Geophys. Astrophys. Fluid Dyn. 39, 57 (1997). S. Rionero ,J.Math.Anal.Appl. ( t o a p p e a r ) , (2004). S. Rionero, Math.Bio.Eng. 3, n . 1 189 (2006). S. Rionero: "Functional for the coincidence between linear and nonlinear stability with applications to spatial ecology and double diffusive convection". In: Proc. Waves and Stability in Continuous Media, WASCOM 2005, Acireale, June 2005 (to appear). S. Rionero: "Z -stability of the solutions to a nonlinear binary reactiondiffusion system of P.D.Es." Rend. Accademia dei Lincei. Fasc.4, 2005 (To appear). S. Rionero: "Global nonlinear i n s t a b i l i t y for double diffusive convection in porous media with and without rotation". (To appear). B. Straughan, Dynamics of Atmosphere and Oceans 35, 351 (2002). G. Veronis, Astrophys. J. 137, 641 (1963) .
O N P E N E T R A T I V E C O N V E C T I O N IN P O R O U S M E D I A D R I V E N B Y Q U A D R A T I C SOURCES
F . C A P O N E , M. G E N T I L E A N D S. R I O N E R O University of Naples Federico II Department of Mathematics and Applications "R. Caccioppoli" Complesso Universitario Monte S. Angelo - Via Cinzia, 80126 Naples E-mail:
[email protected],
[email protected],
[email protected] ITALY
The penetrative convection in a horizontal binary fluid mixture layer with an internal heating due to a quadratic concentration source, according to the Darcy Oberbeque - Boussinesq model, is considered. Conditions guaranteeing the onset of penetrative convection are obtained.
1. Introduction Let E be a horizontal porous layer saturated by a binary fluid mixture and bounded by the planes LTi, II2 from below and above, kept at constant temperatures concentrations. In the present paper, in the framework of Darcy-Boussinesq model, we study the onset of penetrative convection when a quadratic heat source, depending on concentration, is distributed in the layer. In Section 2 the mathematical model and the perturbation equations to the motionless state ms are introduced. Finally, in Section 3, we study the ms density profile, localize the extrema and - following the methodology of Rionero 5 - 9 - conditions for the onset of penetrative convection are found. 2. Basic equations and steady state solution Let us consider a homogeneous fluid mixture contained in a horizontal infinite layer S = {(x,y,z) e R 3 : 0 < z < d} of a porous medium, under the action of a vertical gravity field g = — gk and a distributed heat source. We assume that the Oberbeck-Boussinesq approximation is valid and that the flow in the porous medium is governed by the Darcy's law, i.e. 1'i
Vp=-^v-p0[l-aT(T-T0) fc
+ ac(C-Co)}gk,
Vv =0 I1)
„
2
ATt + v • VT = kTAT + PC , 83
Tu, CL > Cy(> 0). Equations (l)-(2) admit the motionless state ms = {0,Ts,ps,Cs} given by: Vp s = -Pog[l
T.{z) = -P
- aT(Ts - T0) + ac(Cs - C 0 )]k; Cs{z) = -SC Z- + CL z ^ 4£ +6£ +3£ +2e )Z a
i- 4 €-^ " ^ d
with /?* = ^ g ^
> 0,£1
= g(>
1),
£2
= g
(3)
e [0,1], SC = CL -
Cu(> 0), r = Tu - TL(< 0). We denote by {u = (u,v,w), 9, T, ir} the perturbations to the (seepage) velocity, temperature, concentration and pressure fields, respectively. On introducing the dimensionless quantities
R
_ .
\gaTkd\T\ ukr
V
vk-T
where RT and Re are thermal and solute Rayleigh numbers, respectively. Dropping all asterisks, the dimensionless equations in IR2 x [0,1] are: ' VTT = - u + RT6k - RcTk, V •u = 0 1 + l}w + A9 + 2^VLeN(ex 1 + u • V6> = RT[jF(z)
- z)T+
+ 12 7 ^V
(4)
Re
^ e Le r,t + ie u • v r = i?c™ + Ar. in which 7 = —-, e = — (reduced porosity) and F(z) = 4z 3 - 12eiz 2 + 12£?z - {e\ + 3e? + 2eie 2 ) • a
The subscripts / and m refer, respectively, to the fluid and to the mixture.
(5)
85
To (4) we append the stress free boundary conditions w = 9 = T = 0on z = 0,1. We assume that the perturbation fields are periodic in the x and y directions of periods 2n/ax, 2ir/ay and denote by Q, = [0,2Tr/ax] x [0,2ir/ay} x [0,1] the periodicity cell, a = (a2 + a2)1/2 the wave number and by < • > and || • || the L 2 (fi)-scalar product and the L 2 (fi)-norm, respectively. Finally to ensure that the steady state (3) is unique, we assume that < u > = < v > = 0. 3. Onset of penetrative convection By virtue of p(T, C) = po[l — arT + acC] in view of (3), setting z* = z/d, on ms it turns out that p(Ts,Cs)
= P0aTp*[z*4 - 4elZ*3 + 6e2z*2 - (e§ + 3e? + 2e1e2)z*]+ -po{aTT + ac5C)z* + p0[l - aTTL + acCL].
[
'
Dropping the stars and setting p3(z) = p(Ts,Cs), since f3* > 0, p's{z) is an increasing function and it turns out that if p's(0) < 0 and p's(l) > 0, i.e. 0 < \(3*\(e22 + 3e 2 + 2eie 2 ) + r + ^5C
< 4|/F| (1 - 3ex + 3e2)
(7)
ay
then 3!zi G (0,1) : ps{z\) = mmps(z).
Denoting by IT3 the inner plane
z = z\, S is divided into the two parts (IIi, II3) and (II3, II2), "potentially stable" and "potentially unstable", respectively. The penetrative convection occurs only when ms is unstable 10 ' 11 . Therefore its onset is guaranteed by ms instability conditions. On following the methodology introduced by Rionero 5 - 9 , setting am(z) =
ffj
2
a2n(z) = - ^
^
[1 + jF(z)} - ( n V + a2) [1 + lF(z)) + 2*yy/UN(£l
a2 RT RC eLe(n2ir2 + a2) '
__ "
- z)
a2 R2C eLe(n2TT2 + a2)
(8) n2it2 + a2 sLe
with a procedure completely analogous to the which one of Section 5 of [12], one obtains f dS
™
^(
0 1
A + a * I \ ) - U m - VSW + 1 2
m
t.
at
-'ST(n„.a.±n.,r.\-Ln...
1
7
^ ^
[S^f
(9) r
. \7
