The method of intrinsic scaling: systematic approach to regularity for degenerate and singular PDEs

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1930

José Miguel Urbano

The Method of Intrinsic Scaling A Systematic Approach to Regularity for Degenerate and Singular PDEs

123

Author José Miguel Urbano CMUC, Department of Mathematics University of Coimbra 3001-454 Coimbra Portugal [email protected]

ISBN: 978-3-540-75931-7 e-ISBN: 978-3-540-75932-4 DOI: 10.1007/978-3-540-75932-4 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921371 Mathematics Subject Classification (2000): 35D10, 35K65 c 2008 Springer-Verlag Berlin Heidelberg ° 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com

To Martim, Xavier and Catarina.

Preface

When I started giving talks on regularity theory for degenerate and singular parabolic equations, a fixed-point in the conversation during the coffeebreak that usually followed the seminar was the apparent contrast between the beauty of the subject and its technical difficulty. I could not agree more on the beauty part but, most of the times, overwhelmingly failed to convince my audience that the technicalities were not all that hard to follow. As in many other instances, it was the fact that the results in the literature were eventually stated and proved in their most possible generality that made the whole subject seem inexpugnable. So when I had the chance of preparing a short course on the method of intrinsic scaling, I decided to present the theory from scratch for the simplest model case of the degenerate p-Laplace equation and to leave aside technical refinements needed to deal with more general situations. The first part of the notes you are about to read is the result of that effort: an introductory and self-contained approach to intrinsic scaling, aiming at bringing to light what is really essential in this powerful tool in the analysis of degenerate and singular equations. As another striking feature of the method is its pervasiveness in terms of the applications, in the second part of the book, intrinsic scaling is applied to several models arising from flows in porous media, chemotaxis and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for an important and relevant class of nonlinear partial differential equations. The analysis of degenerate and singular parabolic equations is an extremely vast and active research topic and in this contribution there is, by no means, any intention to exhaust the theory. On the contrary, the focus is on a particular subject – the (H¨ older) continuity of solutions – and a unifying set of ideas. We hope that the careful study of theses notes will enable the reader to master the essential features of the method of intrinsic scaling, which is instrumental in dealing with more elaborate aspects of the theory, like the boundedness of solutions, Harnack inequalities or systems of equations.

VIII

Preface

The first four chapters contain material that would fit well in an advanced graduate course on regularity theory for partial differential equations. Each chapter corresponds roughly, with the exception of the first one, to two 90 min. classes. Chapters 5–7 are independent from one another and each could be chosen to complement the course, according to individual preferences. I would probably suggest choosing chapter 5 for that purpose. These lecture notes had its origin in a minicourse I delivered at the 2005 Summer Program of IMPA in Rio de Janeiro. Later that year, I taught a shorter version of the course at the University of Florence. I would like to thank Marcelo Viana and Vincenzo Vespri for their kind invitations and for the wonderful hospitality. I am also indebted to all the colleagues and students who took the course for their interest and input and, in particular, to my former PhD student Eurica Henriques. Finally, I warmly thank Emmanuele DiBenedetto for his continuing support and advice. Coimbra, November 2007

Jos´e Miguel Urbano

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I The Method of Intrinsic Scaling 2

Weak Solutions and a Priori Estimates . . . . . . . . . . . . . . . . . . . . 2.1 Definition of Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Local Energy Estimates: The Building Blocks of the Theory . . 2.3 Local Logarithmic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Some Technical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 15 17

3

The Geometric Setting and an Alternative . . . . . . . . . . . . . . . . . 3.1 A Geometry for the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The First Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Role of the Logarithmic Estimates: Expansion in Time . . . 3.4 Reduction of the Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 25 28 31

4

Towards the H¨ older Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Expanding in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Reducing the Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Defining the Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Recursive Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 39 41 44 47

Part II Some Applications 5

Immiscible Fluids and Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Flow of Two Immiscible Fluids through a Porous Medium 5.2 Rescaled Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Focusing on One Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 53 55

X

Contents

5.4 Behaviour Near the other Degeneracy . . . . . . . . . . . . . . . . . . . . . . 67 5.5 A Problem in Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6

Flows in Porous Media: The Variable Exponent Case . . . . . . 6.1 The Porous Medium Equation in its Own Geometry . . . . . . . . . 6.2 Reducing the Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Analysis of the Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 89 95

7

Phase Transitions: The Doubly Singular Stefan Problem . . . 107 7.1 Regularization of the Maximal Monotone Graph . . . . . . . . . . . . . 108 7.2 A Third Power in the Energy Estimates . . . . . . . . . . . . . . . . . . . . 110 7.3 The Intrinsic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Analyzing the Singularity in Time . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.5 The Effect of the Singularity in the Principal Part . . . . . . . . . . . 126 7.5.1 An Equation in Dimensionless Form . . . . . . . . . . . . . . . . . 130 7.5.2 Expansion in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

1 Introduction

Many relevant phenomena, not only in the natural sciences but also in engineering and economics, are modeled by (systems of) partial differential equations (PDEs) that exhibit some sort of degeneracy or singularity. Examples include the motion of multi-phase fluids in porous media, the melting of crushed ice (and phase transitions, in general), the behavior of composite materials or the pricing of assets in financial markets. Because of its significance in terms of the applications, but also due to the novel analytical techniques that it generates, the class of degenerate and singular parabolic equations is an important branch in the contemporary analysis of partial differential equations. A central example, that will serve as a prototype along the text, is the parabolic p−Laplace equation ut − div |∇u|p−2 ∇u = 0,

p > 1.

If p > 2, the equation is degenerate in the space part since its modulus of ellipticity |∇u|p−2 vanishes at points where |∇u| = 0. If 1 < p < 2, the modulus of ellipticity becomes unbounded at points where |∇u| = 0 and the equation is said to be singular . In general, the nature and origin of the degeneracy or singularity may be quite different but it produces a common effect in the equation, namely the weakening of its structure and the prospect that some of the properties of its solutions be lost. From the point of view of regularity theory, the challenge is to understand to what extent this weakening of the structure, at the points where the PDE degenerates or becomes singular, compromises the regularizing effect that is typical of parabolic equations. Results on the continuity of weak solutions are paramount in this context since they assure that no singularities arise as a consequence of the degradation of the structure of the PDE. The purpose of these lecture notes is to describe intrinsic scaling, a method for obtaining continuity results for the weak solutions of degenerate and singular parabolic equations, and to convince the reader of the strength of this

2

1 Introduction

approach to regularity, by giving evidence of its wide applicability in different situations. To understand what is at stake, let us start by placing the problem in its historical context. As in many other mathematical journeys, it all started with one of Hilbert’s problems. Hilbert’s 19th Problem In 1900, in a then obscure and now legendary session of the International Congress of Mathematicians in Paris, David Hilbert presented his list of 23 problems that would shape the mathematics of the newborn century. Two of those problems were related to the Calculus of Variations: 19th. are the solutions of regular problems in the Calculus of Variations always necessarily analytic? 20th. do regular problems in the Calculus of Variations always possess a solution, (...) extending, if need be, the notion of solution? To be specific, suppose we want to minimize the functional
f [∇w(x)] dx, I[w] =

(1.1)

Ω

over all w : Ω → R such that w = g on ∂Ω. Here, Ω ⊂ Rd and f : Rd → R is a given real function (possibly nonlinear), the so-called Lagrangian. The minimization problem is said to be regular if the Lagrangian f (ζ) is regular and convex. The problem of minimizing the functional (1.1) is associated with its Euler-Lagrange equation (fζi (∇u))xi = 0 in Ω, solved by any minimizer. If one takes the Euler-Lagrange equation for a minimizer w∗ and differentiate it with respect to xk , the result is that, for any k ∈ {1, 2, . . . , d}, the partial derivative (w∗ )xk := vk solves the linear secondorder partial differential equation   aij (x) uxj x = 0 in Ω, (1.2) i

with coefficients aij (x) := fζi ζj (∇w∗ (x)). The equation is (uniformly) elliptic since f is (strictly) convex. The regularity problem for minimizers of convex functionals is thus closely related to the problem of the regularity of the solutions of elliptic equations. In 1904, Bernstein proved that a C 3 solution of equation (1.2) is necessarily analytic, what was seen at the time as a solution to the problem. The subsequent progress focused on weakening the departing regularity required to derive the analyticity. All the results obtained (by E. Hopf, Morrey and Stampacchia, among others) were based, one way or the other, in perturbation

1 Introduction

3

arguments and the comparison of solutions with harmonic functions. As such, they all required some sort of regularity of the coefficients aij (at least continuity). Without the regularity of the coefficients – assuming, for example, they were merely bounded – it was not possible to establish the regularity of the solutions of equation (1.2), let alone that of the minimizers of (1.1). The available theory was based on Schauder estimates that, to simplify, placed the solutions of (1.2) in C k+1,α if the coefficients aij belonged to C k,α . This supported the following iterative reasoning (known as bootstrap argument) w∗ ∈ C 1,α ⇒ aij := fζi ζj (∇w∗ ) ∈ C 0,α ⇒ vk ∈ C 1,α ⇓ vk ∈ C 2,α ⇐ aij := fζi ζj (∇w∗ ) ∈ C 1,α ⇐ w∗ ∈ C 2,α ⇓ w∗ ∈ C 3,α ⇒

...

so ultimately w∗ ∈ C ∞ . The analyticity then followed. Meanwhile, the existence theory for the Calculus of Variations started developing through the use of direct methods. Imposing, in addition to the convexity, a coercivity assumption on the Lagrangian (of the type |f (ζ)| ≥ δ |ζ|2 − β, ζ ∈ Rd ), it was possible to guarantee the existence of a minimizer, provided that the notion of solution was adequately extended – which was entirely in the spirit of the formulation of Hilbert’s 20th problem. The appropriate class of admissible functions, for the corresponding minimization problem, turned out to be the Sobolev space H 1 , i.e., the space of L2 functions whose first derivatives (in the sense of distributions) still belong to L2 . There was thus a gap between the existence and regularity classes. Extending the notion of solution, it was possible to obtain the existence of a minimizer in H 1 . From this, the measurability and, with natural hypothesis on the Lagrangian (|D2 f (ζ)| ≤ C, ζ ∈ Rd ), the boundedness of the aij followed. The question, in the early 1950’s, boiled down to showing that the solutions of (1.2), with coefficients merely measurable and bounded, were H¨older continuous: ?

w∗ ∈ H 1 ⇒ aij := fζi ζj (∇w∗ ) ∈ L∞ ⇒ vk ∈ C 0,α ⇒ w∗ ∈ C 1,α It was, arguably, the most important problem in the analysis of partial differential equations at that time. The problem had been solved by Morrey, in 1938 [41], but only for d = 2, and the techniques used were typically bidimensional, involving complex analysis and quasi-conformal mappings. In the geral case, only Stampacchia’s estimates were known, giving u ∈ H 2 ; the problem remained widely open. Enters De Giorgi In 1956, Ennio De Giorgi, a young mathematician of Pisa, then twenty-nine, submits to the Rendiconti dell’Accademia Nazionale dei Lincei a four pages

4

1 Introduction

note, titled Sull’analiticit` a degli estremali degli integrali multipli, announcing the proof (later published in [10]) of the H¨ older continuity of the functions that satisfy certain integral conditions – later known as elements of De Giorgi’s class. As the solutions of (1.2), with measurable and bounded coefficients, belong to that class, a positive answer to Hilbert’s 19th problem, in the scalar case and any dimension, followed as an immediate consequence. In the vectorial case, the answer is negative and it was again De Giorgi, around ten years later, who constructed a counter-example of a discontinuous solution to a regular (in the sense of Hilbert) variational system. De Giorgi uses a completely original approach to obtain the a priori that lead to the solution of the problem. The following is Bombieri’s testimony [5] (translated freely from the Italian): Once I asked De Giorgi how he got to the idea that led him to solve this problem. He replied as if it was all an indirect consequence of another problem, much more difficult, that he was studying at that moment, namely the isoperimeteric problem in several dimensions, and started explaining the connections between the two problems. I then realized that De Giorgi looked at these functions of several variables literally as geometric objects in space. In his explanation, he kept moving his hands as if he was touching an invisible surface, and showing how to perform his operations and transformations, cutting and pasting invisible masses from one side to the other, leveling and filling the peaks and valleys of theses surfaces. In the specific case, it consisted of taking the level curves of the surface that solved the problem and applying his isoperimetric property. To me, it was an usual way of doing analysis, a field that often requires the use of rather fine estimates, that the normal mathematician grasps more easily through the formulas than through the geometry.

The work of De Giorgi concerns uniformly elliptic linear equations but the linearity does not play a role in the proofs. This allowed Ladyzhenskaya and Uralt’seva [38] to extend, in the mid 1960’s, the results on the H¨ older continuity of weak solutions to quasi-linear equations of the form div a(x, u, ∇u) = 0

in Ω,

(1.3)

with structure assumptions of the type ⎧ ⎨ a(x, u, ∇u) · ∇u ≥ C0 |∇u|p − C ⎩

  |a(x, u, ∇u)| ≤ C |∇u|p−1 + 1 ,

(1.4)

with p > 1, and constants C0 > 0 and C ≥ 0. The generalization is twofold: the principal part a(x, u, ∇u) is permitted to have a nonlinear dependence in ∇u, and a non-linear growth with respect to |∇u|. The latter is of particular interest since equation (1.3) might be either degenerate or singular, as illustrated by the archetypal p−Laplace equation div |∇u|p−2 ∇u = 0.

1 Introduction

5

From Elliptic to Parabolic Let us now consider the parabolic analogues of equations (1.2) and (1.3), that is, with ΩT = Ω × (0, T ], 0 < T < ∞,   (1.5) ut − aij (x, t) uxj x = 0 in ΩT , i

with bounded and measurable coefficients aij satisfying an ellipticity condition, and ut − div a(x, t, u, ∇u) = 0

in ΩT ,

(1.6)

with a satisfying structure assumptions analogous to (1.4). Moser [43] proved that weak solutions of (1.5) are locally H¨ older continuous in ΩT . Since the linearity is immaterial to the proof, one might expect, as in the elliptic case, an extension of these results to quasi-linear equations of the type (1.6), where the structure condition is as in (1.4). Surprisingly though, the methods of De Giorgi and Moser could not be extended. Ladyzhenskaya et als. [37] proved that solutions of (1.6) are H¨ older continuous, provided the principal part has exactly a linear growth with respect to |∇u|, i.e., if p = 2 in the structure assumptions corresponding to (1.4). Analogous results were established by Kruzkov [33, 34] and by Nash [44] using entirely different methods. Thus it appears that unlike the elliptic case, the degeneracy or singularity of the principal part plays a peculiar role, and for example, for the parabolic p−Laplace equation ut − div |∇u|p−2 ∇u = 0 one could not establish whether a solution is locally H¨ older continuous. Intrinsic Scaling: A New Approach to Regularity The issue remained open until the mid 1980’s when DiBenedetto [12] showed that the solutions of general quasilinear equations of the type of (1.6) are locally H¨ older continuous for p > 2. In the early 1990’s, the theory was extended [9] to include also the case 1 < p < 2. Surprisingly, the same techniques could be suitably modified to establish the local H¨ older continuity of any local solution of quasilinear porous medium-type equations. These modified methods, in turn, were crucial in proving that weak solutions of the p−Laplace equation 1,α . are of class Cloc These results follow, one way or another, from the single unifying idea of intrinsic scaling: the diffusion processes in the equations evolve in a time scale determined instant by instant by the solution itself, so that, loosely speaking, they can be regarded as the heat equation in their own intrinsic timeconfiguration. A precise description of this fact, as well as its effectiveness, is linked to its technical implementation, which we will develop in this set of lecture notes.

6

1 Introduction

The continuity of a solution at a point follows from measuring its oscillation in a sequence of nested and shrinking cylinders, with vertex at that point, and showing that the oscillation converges to zero as the cylinders shrink to the point. When it is possible to describe quantitatively how the oscillation converges, a modulus of continuity is derived. The idea behind the method of intrinsic scaling is to perform this iterative process in cylinders that reflect the structure of the equation. By this, we mean cylinders whose dimensions, with respect to the standard parabolic cylinders, are redefined in terms of scaling factors that take into account the nature of the degeneracy or singularity, and depend on the oscillation of the solution itself (thus, the term intrinsic). The cylinders should trivially reduce to the standard parabolic cylinders, reflecting the natural homogeneity between the space and time variables for the heat equation, in the particular case of a uniformly parabolic equation. The punch line of the theory is that the equation behaves, in its own geometry, like the heat equation. The building blocks of the method of intrinsic scaling are a priori estimates for the weak solutions of the equation. Actually, there is more to it than that. Once these estimates are obtained, we can forget the equation and the problem becomes, purely, a problem in analysis: showing that functions that satisfy certain integral inequalities belong to a certain regularity class (e.g., are locally H¨ older continuous). It does not really matter if these functions are solutions of an equation or extremals of a functional in the Calculus of Variations or neither of those; what counts is that they satisfy the integral estimates. These integral inequalities on level sets measure the behaviour of the function near its infimum and its supremum in the interior of the cylinder. In the case of solutions of degenerate or singular equations, these estimates are not homogeneous since they involve integral norms corresponding to different powers. This lack of homogeneity precludes the use of certain functional inclusions because the appropriate norms are not disclosed in the analysis. Through intrinsic scaling, we are able to recover the homogeneity in the estimates, once we rewrite them over the intrinsically rescaled cylinders. The difficulty in the analysis is thus absorbed by the geometry. Emerging Trends In a series of papers [17, 18, 19], DiBenedetto, Gianazza and Vespri established recently an intrinsic Harnack inequality for nonnegative solutions of degenerate and singular parabolic PDEs with the full quasi-linear structure, a problem that remained open for about 40 years. The result, that provides an alternative and independent proof of the H¨ older continuity of solutions, was known only for the particular case of the parabolic p-Laplacian, and the available proof was clearly of a restricted applicability since it used, in an essential way, the maximum principle and comparison with p-heat potentials. In a way, this new result parallels the celebrated work [43] of Moser in the 1960’s, when the Harnack inequality for general nondegenerate (p = 2) quasilinear parabolic

1 Introduction

7

PDEs was obtained. The true advance in Moser’s proof was also in bypassing the use of potentials that were essential in the approach of Hadamard [25] and Pini [45] concerning the heat equation. There is even a further merit in the contribution of DiBenedetto et als: unlike Moser’s proof, no use is made of some fine analytical properties of BMO spaces (the use of John-Nirenberg lemma was rather complicated in the original paper of Moser, who provided a simpler proof some time later). The arguments are measure-theoretical in nature and, as the authors remark, hold the promise of a wider applicability. See also [36] for a different perspective in the degenerate case. For more on contemporary issues related to regularity estimates for degenerate and singular elliptic and parabolic equations, we address the interested reader to the recent special issue [20], where several contributions are collected. Outline of the Lecture Notes We have decided to present the theory for the model case of the degenerate p−Laplace equation to bring to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations. In chapter 2, the precise definition of weak solution is introduced for the model problem, and we derive local energy and logarithmic estimates, the building blocks of the theory. Chapter 3 deals with the construction of the appropriate geometric setting, bringing about in full detail the idea of intrinsic scaling; it also highlights the precise role of the logarithmic estimates. The 4th chapter culminates with the proof of the H¨ older continuity, after the analysis of an alternative aimed at reducing the oscillation of the solution. The last section contains a series of generalizations, mainly to equations with the full quasilinear structure and of porous medium type. This first part is essentially an edited version of the second section of [21]; some proofs are presented in more detail but the overlapping of the material is substantial. The division into sections and the order in which the issues are treated is rather different though, and it is intended to further simplify the reading. The second part is devoted to a series of three applications of the theory to relevant models arising from flows in porous media, chemotaxis and phase transitions. In chapter 5, the flow of two immiscible fluids through a porous medium is studied and intrinsic scaling is used to obtain the H¨ older continuity of the saturation, that satisfies a PDE with a two-sided degeneracy. The same type of structure arises in a model of chemotaxis with volume–filling effect and the extension to this case, which basically consists in dealing appropriately with an extra lower order term, is also included. In chapter 6, we obtain the continuity of the weak solutions of the porous medium equation with a variable exponent, generalizing the classical result to an increasingly popular context. Finally, in chapter 7, the Stefan problem for the singular p-Laplacian

8

1 Introduction

is considered and intrinsic scaling is again used to derive the continuity of the temperature, showing that no jumps occur across the free boundary. These three examples were originally treated in [53, 4], [28] and [27], respectively. The contents of the chapters reflect this fact, with simplifications and extra remarks whenever appropriate. The novelty here lies in the unified approach, intended to convince the reader of the strength of the method of intrinsic scaling as a systematic approach to regularity for degenerate and singular PDEs.

2 Weak Solutions and a Priori Estimates

We will concentrate on the parabolic p−Laplace equation ut − div |∇u|p−2 ∇u = 0 ,

p > 1,

(2.1)

a quasilinear second-order partial differential equation, with principal part in divergence form. If p > 2, the equation is degenerate in the space part, due to the vanishing of its modulus of ellipticity |∇u|p−2 at points where |∇u| = 0. The singular case corresponds to 1 < p < 2: the modulus of ellipticity becomes unbounded at points where |∇u| = 0. In this chapter we place no restriction on the values of p > 1. The theory is markedly different in the degenerate and singular cases and we will later restrict our attention to p > 2. The results extend to a variety of equations and, in particular, to equations with general principal parts satisfying appropriate structure assumptions and with lower order terms. We have chosen to present the results and the proofs for the particular model case (2.1) to bring to light what we feel are the essential features of the theory. Remarks on generalizations, which in some way or another correspond to more or less sophisticated technical improvements, are left to a later section.

2.1 Definition of Weak Solution Let Ω be a bounded domain in Rd , with smooth boundary ∂Ω. Let ΩT = Ω × (0, T ] ,

T > 0,

be the space-time domain, with lateral boundary Σ = ∂Ω × (0, T ) and parabolic boundary ∂p ΩT = Σ ∪ (Ω × {0}) . We start with the precise definition of local weak solution for (2.1).

12

2 Weak Solutions and a Priori Estimates

Definition 2.1. A local weak solution of (2.1) is a measurable function     1,p (Ω) u ∈ Cloc 0, T ; L2loc (Ω) ∩ Lploc 0, T ; Wloc such that, for every compact K ⊂ Ω and for every subinterval [t1 , t2 ] of (0, T ], t2
t2



 −uϕt + |∇u|p−2 ∇u · ∇ϕ dx dt = 0, uϕ dx + (2.2) K

t1

t1

K

    1 0, T ; L2 (K) ∩ Lploc 0, T ; W01,p (K) . for all ϕ ∈ Hloc It would be technically convenient to have at hand a formulation of weak solution involving the time derivative ut . Unfortunately, solutions of (2.1), whenever they exist, possess a modest degree of time-regularity and, in general, ut has a meaning only in the sense of distributions. To overcome this limitation, we introduce the Steklov average of a function v ∈ L1 (ΩT ), defined, for 0 < h < T , by ⎧
t+h ⎪ ⎪ ⎨ h1 v(·, τ ) dτ if t ∈ (0, T − h] t (2.3) vh := ⎪ ⎪ ⎩ 0 if t ∈ (T − h, T ]. The proof of the following lemma follows from the general theory of Lp spaces. Lemma 2.2. If v ∈ Lq,r (ΩT ) then, as h → 0, the Steklov average vh converges to v in Lq,r (ΩT − ), for every ∈ (0, T ). If v ∈ C (0, T ; Lq (Ω)) then, as h → 0, the Steklov average vh (·, t) converges to v(·, t) in Lq (Ω), for every t ∈ (0, T − ) and every ∈ (0, T ). It is a simple exercise to show that the definition of local weak solution previously introduced is equivalent to the following one. Definition 2.3. A local weak solution of (2.1) is a measurable function     1,p (Ω) u ∈ Cloc 0, T ; L2loc (Ω) ∩ Lploc 0, T ; Wloc such that, for every compact K ⊂ Ω and for every 0 < t < T − h,


   (uh )t ϕ + |∇u|p−2 ∇u h · ∇ϕ dx = 0,

(2.4)

K×{t}

for all ϕ ∈ W01,p (K). We will show that locally bounded solutions of (2.1) are locally H¨ older continuous within their domain of definition. No specific boundary or initial values need to be prescribed for u. A theory of boundedness of weak solutions

2.2 Local Energy Estimates: The Building Blocks of the Theory

13

of (2.1) is quite different from the linear theory (cf. [14]): weak solutions are locally bounded only if d(p − 2) + p > 0. It can be shown by counterexample that this condition is sharp. Although the arguments below are of local nature, to simplify the presentation we assume that u is a.e. defined and bounded in ΩT and set M := u L∞ (ΩT ) .

2.2 Local Energy Estimates: The Building Blocks of the Theory The building blocks of the method of intrinsic scaling are a priori estimates for weak solutions. Once these estimates are obtained, we can forget the equation and the problem becomes, purely, a problem in analysis: showing that functions that satisfy certain integral inequalities belong to a certain regularity class (e.g., are locally H¨ older continuous). These estimates are integral inequalities on level sets that measure the behaviour of the function near its infimum and its supremum in the interior of an appropriate cylinder. Given a point x0 ∈ Rd , denote by Kρ (x0 ) the d-dimensional cube with centre at x0 and wedge 2ρ:

 Kρ (x0 ) := x ∈ Rd : max |xi − x0 i | < ρ 1≤i≤d

and put Kρ := Kρ (0); given a point (x0 , t0 ) ∈ Rd+1 , the cylinder of radius ρ and height τ > 0 with vertex at (x0 , t0 ) is (x0 , t0 ) + Q(τ, ρ) := Kρ (x0 ) × (t0 − τ, t0 ). (x0 , t0 ) •

ρ

τ

2 Weak Solutions and a Priori Estimates

14

We write Q(τ, ρ) to denote (0, 0) + Q(τ, ρ). We use the usual notations for the positive and negative parts of a function: v+ = max(v, 0)

and

v− = (−v)+ .

We now deduce the energy estimates. Without loss of generality, we restrict to cylinders with vertex at the origin (0, 0), the changes being obvious for cylinders with vertex at a generic (x0 , t0 ). Consider a cylinder Q(τ, ρ) ⊂ ΩT and let 0 ≤ ζ ≤ 1 be a piecewise smooth cutoff function in Q(τ, ρ) such that |∇ζ| < ∞

and

ζ(x, t) = 0 ,

x ∈ Kρ .

(2.5)

Proposition 2.4. Let u be a local weak solution of (2.1) and k ∈ R. There exists a constant C ≡ C(p) > 0 such that, for every cylinder Q(τ, ρ) ⊂ ΩT ,





sup

−τ

b ± (a + c)

b±c

whose first derivative is ⎧ 1 < ⎪ ⎪ ⎨ (b − s) ± (a + c) if b ± c > s   ± ψ{a,b,c} (s) = ⎪ ⎪ < ⎩ 0 if s > b ± c

< >

b ± (a + c) ≥ ≤

0,

16

2 Weak Solutions and a Priori Estimates

and second derivative, off s = b ± c, is 

± ψ{a,b,c}



  2 ± = ψ{a,b,c} ≥ 0.

Now, given a bounded function u in a cylinder (x0 , t0 ) + Q(τ, ρ) and a number k, define the constant ± := Hu,k

ess sup (x0 ,t0 )+Q(τ,ρ)

|(u − k)± | .

The following function was introduced in [11] and since then has been used as a recurrent tool in the proof of results concerning the local behaviour of solutions of degenerate and singular equations:   ± ± , (u − k)± , c ≡ ψ ±H ± ,k,c (u) , . (2.7) 0 < c < Hu,k Ψ ± Hu,k { u,k } From now on, when referring to this function we will write it as ψ ± (u), omitting the subscripts, whose meaning will be clear from the context. Let x → ζ(x) be a time-independent cutoff function in Kρ (x0 ) satisfying (2.5). The logarithmic estimates in cylinders Q(τ, ρ) with vertex at the origin read as follows. Proposition 2.6. Let u be a local weak solution of (2.1) and k ∈ R. There exists a constant C ≡ C(p) > 0 such that, for every cylinder Q(τ, ρ) ⊂ ΩT ,

 ± 2 p  ± 2 p sup ψ (u) ζ dx ≤ ψ (u) ζ dx −τ k2 } , [v < k1 ] := {x ∈ Ω : v(x) < k1 } , [k1 < v < k2 ] := {x ∈ Ω : k1 < v(x) < k2 } .

(2.9)

18

2 Weak Solutions and a Priori Estimates

Lemma 2.7 (De Giorgi, [10]). Let v ∈ W 1,1 (Bρ (x0 )) ∩ C (Bρ (x0 )), with ρ > 0 and x0 ∈ Rd , and let k1 < k2 ∈ R. There exists a constant C, depending only on d (and thus independent of ρ, x0 , v, k1 and k2 ), such that
ρd+1 |∇v| dx. (k2 − k1 ) |[v > k2 ]| ≤ C |[v < k1 ]| [k1 0 are given. If X0 ≤ C −1/α b−1/α

2

then Xn → 0 as n → ∞. Lemma 2.10. Let (Xn ) and (Zn ), n = 0, 1, 2, . . ., be sequences of positive real numbers satisfying the recurrence relations ⎧   ⎨ Xn+1 ≤ C bn Xn1+α + Xnα Zn1+κ ⎩

  Zn+1 ≤ C bn Xn + Zn1+κ

where C, b > 1 and α, κ > 0 are given. If X0 + Z01+κ ≤ (2C)−

1+κ σ

then Xn , Zn → 0 as n → ∞.

b−

1+κ σ2

,

with

σ = min{α, κ},

2.4 Some Technical Tools

19

3. An Embedding Theorem Let V0p (ΩT ) denote the space   V0p (ΩT ) = L∞ (0, T ; Lp (Ω)) ∩ Lp 0, T ; W01,p (Ω) endowed with the norm

u pV p (ΩT ) = ess sup u(·, t) pp,Ω + ∇u pp,ΩT . 0≤t≤T

The following embedding theorem holds (cf. [14, page 9]). Theorem 2.11. Let p > 1. There exists a constant γ, depending only on d and p, such that for every v ∈ V0p (ΩT ), p

v pp,ΩT ≤ γ | |v| > 0 | d+p v pV p (ΩT ) . 4. A Poincar´ e-type Inequality Let Ω ⊂ Rd be a bounded and convex set. Consider a function ϕ ∈ C(Ω), 0 ≤ ϕ ≤ 1, such that the sets [ϕ > k],

0 1 is to be fixed later depending only on the data (see (4.15)). Note that for p = 2, i.e., in the non-degenerate case, these are the standard parabolic cylinders reflecting the natural homogeneity between the space and time variables. We will assume, without loss of generality, that R
λ, p−2

and assume that N0 =

a0 = θ



ω 2λ ω 2

2−p = 2 (λ−1)(p−2)

(3.7)

is an integer. Thus, we consider Q(a0 Rp , R) as being divided in subcylinders, all alike and congruent with Q(θRp , R): (0, 0) •

(0, t∗) • θRp (0, t∗ ) + Q(θRp , R)

Q(a0 Rp , R)

The proof of the H¨ older continuity of a weak solution u now follows from the analysis of two complementary cases. We briefly describe them in the

3.2 The First Alternative

25

following way: in the first case we assume that there is a cylinder of the type (0, t∗ )+Q(θRp , R) where u is essentially away from its infimum. We show that going down to a smaller cylinder the oscillation decreases by a small factor that we can exhibit. If that cylinder can not be found then u is essentially away from its supremum in all cylinders of that type and we can compound this information to reach the same conclusion as in the previous case. We state this in a precise way. For a constant ν0 ∈ (0, 1), that will be determined depending only on the data, either The First Alternative: there is a cylinder of the type (0, t∗ ) + Q(θRp , R) for which

 (x, t) ∈ (0, t∗ ) + Q(θRp , R) : u(x, t) < µ− + ω 2 ≤ ν0 |Q(θRp , R)| or this does not hold. Then, since µ+ −

ω 2

(3.8)

= µ− + ω2 , it holds

The Second Alternative: for every cylinder of the type (0, t∗ ) + Q(θRp , R)

 (x, t) ∈ (0, t∗ ) + Q(θRp , R) : u(x, t) > µ+ − ω 2 < 1 − ν0 . |Q(θRp , R)|

(3.9)

3.2 The First Alternative We start the analysis assuming the first alternative holds. Lemma 3.1. Assume (3.3) is in force. There exists a constant ν0 ∈ (0, 1), depending only on the data, such that if (3.8) holds for some t∗ as in (3.6) then    p ω R R − ∗ . , u(x, t) > µ + , a.e. in (0, t ) + Q θ 4 2 2 Proof. Take the cylinder for which (3.8) holds and assume, by translation, that t∗ = 0. Let R R + n+1 , n = 0, 1, . . . , Rn = 2 2 and construct the family of nested and shrinking cylinders Q(θRnp , Rn ). Consider piecewise smooth cutoff functions 0 ≤ ζn ≤ 1, defined in these cylinders, and satisfying the following set of assumptions:

26

3 The Geometric Setting and an Alternative

  p ζn = 1 in Q θRn+1 , Rn+1 ; |∇ζn | ≤

ζn = 0 on ∂p Q (θRnp , Rn ) ;

2n+1 ; R

2p(n+1) . θRp

0 ≤ (ζn )t ≤

p Observe the family of cylinders starts with Q(θR , R) and converges to   that p Q θ R2 , R2 and that the bounds on the gradient and the time derivative of ζn are strictly related to the dimensions of the cylinders. Write the energy inequality (2.6) over the cylinders Q (θRnp , Rn ), for the functions (u − kn )− , with

kn = µ− +

ω ω + n+2 , 4 2

n = 0, 1, . . . ,

and ζ = ζn . They read, taking into account that ζn vanishes on ∂p Q (θRnp , Rn ),





sup p −θRn k in the cube K R , which implies that 2

 −  ψ (u) (x, − t) = 0 ,

x ∈ KR . 2

Choosing a piecewise smooth cutoff function 0 < ζ(x) ≤ 1, defined in K R and 2

such that ζ = 1 in K R 4

and

|∇ζ| ≤

8 , R

30

3 The Geometric Setting and an Alternative

inequality (2.8) reads


 − 2 p ψ (u) ζ dx

sup

− t µ− + s1 +1 , a.e. in Q  2 8 Proof. Consider the cylinder for which (3.8) holds, let Rn =

R R + n+3 , 8 2

n = 0, 1, . . .

and construct the family of nested and shrinking cylinders Q( t, Rn ), where  t is given by (3.15). Take piecewise smooth cutoff functions 0 < ζn (x) ≤ 1, independent of t, defined in KRn and satisfying |∇ζn | ≤

ζn = 1 in KRn+1 ;

2n+4 . R

Write the local inequalities (2.6) for the functions (u − kn )− , in the   energy cylinders Q  t, Rn , with kn = µ− +

ω ω + s1 +1+n , 2s1 +1 2

n = 0, 1, . . . ,

s1 to be chosen, and ζ = ζn . Observing that, due to Lemma 3.1, we have u(x, − t) > µ− +

ω ≥ kn 4

in K R ⊃ KRn , 2

which implies that (u − kn )− (x, − t) = 0 the estimates read
sup − t µ+ − 2
t∗ − ν20 θRp  ω  ≥ dτ x ∈ KR : u(x, τ ) > µ+ − 2 t∗ −θRp > (1 − ν0 ) |Q(θRp , R)| , which contradicts (4.1).

36

4 Towards the H¨ older Continuity

The next lemma asserts that the set where u(·, t) is close to its supremum is small, not only at the specific time level t◦ , but for all time levels near the top of the cylinder (0, t∗ ) + Q(θRp , R). Lemma 4.1. Assume (4.1) and let (3.3) be in force. There exists s2 ∈ N, depending only on the data, such that    ν 2  ω  0 + |KR | , x ∈ KR : u(x, t) > µ − s2 ≤ 1 − 2 2   for all t ∈ t∗ − ν20 θRp , t∗ . Proof. The proof consists in using the logarithmic inequalities (2.8) applied to the function (u − k)+ in the cylinder KR × (t◦ , t∗ ), with the choices k = µ+ −

ω 2

and

c=

ω 2n+1

,

where n ∈ N will be chosen later. In this cylinder, we have  ω  ω + + ≤ . u − k ≤ Hu,k = ess sup u − µ + 2 + 2 KR ×(t◦ ,t∗ )

(4.3)

+ + If Hu,k ≤ ω4 the result is trivial with the choice s2 = 2. Assuming Hu,k > ω4 , + recall from section 2.3 that the logarithmic function ψ (u) is defined in the whole of KR × (t◦ , t∗ ), and it is given by ⎧   + ⎪ Hu,k ⎪ ω ⎪ if u > k + 2n+1 ⎨ ln + ω H − u + k + + n+1 u,k 2 ψ H + ,k, ω (u) = { u,k 2n+1 } ⎪ ⎪ ⎪ ⎩ ω 0 if u ≤ k + 2n+1 .

From (4.3), we obtain the estimates ψ + (u) ≤ n ln 2

since

+ Hu,k + Hu,k −u+k+

ω 2n+1

≤

ω 2 ω

= 2n ,

(4.4)

2n+1

and

2−p    p−2  ω p−2 +  + (u) = Hu,k −u+k+c ≤ . (4.5) ψ 2 Choosing a piecewise smooth cutoff function 0 < ζ(x) ≤ 1, defined in KR and such that, for some σ ∈ (0, 1), ζ = 1 in K(1−σ)R inequality (2.8) reads

and

|∇ζ| ≤ (σR)−1 ,

4.1 Expanding in Time


sup

t◦ µ+ − n+1 2   2   C 1 − ν0 n + + dσ |KR | , ≤ n−1 1 − ν0 /2 nσ p

for all t ∈ (t◦ , t∗ ). Choose σ so small that dσ ≤ 38 ν02 and n so large that 

n n−1

2

 ν0  (1 + ν0 ) =: β ≤ 1− 2

and

C 3 ≤ ν02 . p nσ 8

(4.7)

Note that β > 1. With this choice of n, the lemma follows with s2 = n + 1.   The same type of conclusion  holds in  an upper portion of the full cylinder Q(a0 Rp , R), say for all t ∈ − a20 Rp , 0 . Indeed, (4.1) holds for all cylinders of the type (0, t∗ ) + Q(θRp , R) so the conclusion of the previous lemma holds true for all time levels ν0 t ≥ −(a0 − θ)Rp − θRp . 2 So if we choose λ such that 2(λ−1)(p−2) ≥ 2 a0 ≥ 2 − ν0 , which is equivalent to θ ν0 a0 −(a0 − θ)Rp − θRp ≤ − Rp . 2 2

we get, recalling (3.7),

(4.8)

4.2 Reducing the Oscillation

39

Corollary 4.2. Assume (4.1) and let (3.3) be in force. Then    ν 2  ω  0 |KR | , x ∈ KR : u(x, t) > µ+ − s2 ≤ 1 − 2 2   for all t ∈ − a20 Rp , 0 .

4.2 Reducing the Oscillation The main result of this chapter states that in fact u is strictly below its supre mum µ+ in a smaller cylinder with the same vertex and axis as Q a20 Rp , R . Proposition 4.3. Assume (4.1) and let (3.3) be in force. The choice of λ can be made so that    p ω a0 R R + . (4.9) , u(x, t) ≤ µ − λ+1 , a.e. in Q 2 2 2 2 Proof. Define R R Rn = + n+1 , n = 0, 1, . . . , 2 2   and construct the family of nested and shrinking cylinders Q a20 Rnp , Rn . Consider piecewise smooth cutoff functions 0 ≤ ζn ≤ 1, defined in these cylinders and satisfying the following set of assumptions:     p ζn = 0 on ∂p Q a20 Rnp , Rn ; , Rn+1 ; ζn = 1 in Q a20 Rn+1 |∇ζn | ≤

2n+1 ; R

0 ≤ (ζn )t ≤

2p(n+1) a0 p . 2 R

The energy inequality (2.6) for the functions (u − kn )+ , with ω ω kn = µ+ − λ+1 − λ+1+n , n = 0, 1, . . . , 2 2   in the cylinders Q a20 Rnp , Rn , and with ζ = ζn , reads


0 p sup (u − kn )2+ ζnp dx + |∇(u − kn )+ ζn | dx dt −

a0 2

p Rn µ+ − σ 2 and  a  ω 0 p R , R : u(x, t) > µ+ − σ . Bσ = (x, t) ∈ Q 2 2 With this notation, (4.12) reads a  0 p R ,R . |Bλ | ≤ ν∗ Q 2 We will use the information contained in Corollary 4.2  to show that this  holds, i.e., that the subset of the cylinder Q a20 Rp , R where u is close to

42

4 Towards the H¨ older Continuity

its supremum µ+ can be made arbitrarily small. Consider the local energy inequalities (2.6) for the functions (u − k)+ in the cylinders Q (a0 Rp , 2R), with ω k = µ+ − s , 2 where s will be chosen later satisfying s2 ≤ s ≤ λ (recall that s2 was chosen in Lemma 4.1). Take a piecewise smooth cutoff function 0 ≤ ζ ≤ 1, defined in Q (a0 Rp , 2R), and such that   ζ = 0 on ∂p Q (a0 Rp , 2R) ; ζ = 1 in Q a20 Rp , R ; |∇ζ| ≤

1 ; R

0 ≤ ζt ≤

2 . a0 Rp

Neglecting the first term on the left hand side of the estimates, we obtain for the indicated choices,

0

0 C p |∇(u − k)+ | dx dt ≤ p (u − k)p+ dx dt a R −a0 Rp K2R − 20 Rp KR
0
C (u − k)2+ dx dt. + a0 Rp −a0 Rp K2R We estimate the two terms on the right hand side of this inequality as follows:
0
C  ω p  a0 p  C p R ,R (u − k) dx dt ≤ Q + Rp −a0 Rp K2R Rp 2s 2 and, recalling the definition of a0 ,
0
C  ω p−2  ω 2  a0 p  C 2 R ,R (u − k) dx dt ≤ Q + a0 Rp −a0 Rp K2R Rp 2λ 2s 2 C  ω p  a0 p  R ,R , ≤ p Q R 2s 2 since s ≤ λ. Gathering results, we reach

C  ω p  a0 p  R ,R . |∇u|p dx dt ≤ p Q R 2s 2 Bs

(4.14)

We next apply Lemma 2.7 to the function u(·, t), for all − a20 Rp ≤ t ≤ 0, and with ω ω ω k2 = µ+ − s+1 , k2 − k1 = s+1 . k1 = µ+ − s , 2 2 2 Observing that, owing to Corollary 4.2,   ν 2 ω  0 |KR | , x ∈ KR : u(x, t) ≤ µ+ − s = |KR | − |Bs (t)| ≥ 2 2

4.3 Defining the Geometry

43

we obtain ω

4C Rd+1 |Bs+1 (t)| ≤ 2 ν0 |KR |




|∇u| dx, 2s+1 Bs (t)\Bs+1 (t)   for t ∈ − a20 Rp , 0 . Integrating over this interval, we conclude that

CR ω |Bs+1 | ≤ 2 |∇u| dx dt 2s+1 ν0 Bs \Bs+1 

 p1 p−1 CR p ≤ 2 |∇u| dx dt |Bs \ Bs+1 | p ν0 Bs   p−1 C ω  a0 p  p1 R , R |Bs \ Bs+1 | p , ≤ 2 Q s ν0 2 2 using also (4.14). Simplifying and taking the

p p−1

power, we obtain

1 a  p−1 p 2p 0 p R ,R |Bs+1 | p−1 ≤ C (ν0 )− p−1 Q |Bs \ Bs+1 | . 2

Since these inequalities are valid for s2 ≤ s ≤ λ, we add them for s = s2 , s2 + 1, s2 + 2, . . . , λ − 1,

  and since the sum on the right hand side is bounded above by Q a20 Rp , R , we obtain p a  p−1 p 2p 0 p R ,R , (λ − s2 ) |Bλ | p−1 ≤ C (ν0 )− p−1 Q 2 that is, |Bλ | ≤

C (λ − s2 )

p−1 p

a  0 p R ,R . (ν0 )−2 Q 2

We obtain (4.12) if λ is chosen so large that C ν02 (λ − s2 ) We finally make the choice  λ = max s2 +



p−1 p

C ν02 ν∗

≤ ν∗ .

p  p−1

1 , 1+ p−2

 (4.15)

(recall that s2 is given through (4.7), ν0 is given by (3.13), and ν∗ is given by (4.12)) thus concluding the proof of the proposition.   Remark 4.4. Observe that the choice of λ was made so that (4.8) holds, and a larger λ is admissible.

44

4 Towards the H¨ older Continuity

Corollary 4.5. Assume (4.1) and let (3.3) be in force. There exists a constant σ1 ∈ (0, 1), depending only on the data, such that u ≤ σ1 ω.

ess osc

Q(

a0 2

p R (R 2 ) , 2 )

Proof. It is similar to the proof of Corollary 3.5 for the choice   1 σ1 = 1 − λ+1 . 2

 

4.4 The Recursive Argument We finally prove the H¨ older continuity of weak solutions through an iterative scheme designed from all previous results. An immediate consequence of Corollaries 3.5 and 4.5 is the following. Proposition 4.6. There exists a constant σ ∈ (0, 1), that depends only on the data, such that ess osc

p R Q(d( R 8 ) , 8 )

u ≤ σ ω.

Proof. Assume (3.3) is in force. Then, by Corollaries 3.5 and 4.5, ess osc

p R Q(d( R 8 ) , 8 )

u ≤ σω, 

since d

R 8

where

p ≤

a0 2



R 2

σ = max{σ0 , σ1 },

(4.16)

p .  

Proposition 4.7. There exists a positive constant C, depending only on the data, such that, defining the sequences Rn = C −n R

ωn = σ n ω,

and

for n = 0, 1, 2, . . . , where σ ∈ (0, 1) is given by (4.16), and constructing the family of cylinders  ω 2−p n with an = , Qn = Q(an Rnp , Rn ) , 2λ where λ > 1 is given by (4.15), we have Qn+1 ⊂ Qn for all n = 0, 1, 2, . . .

and

ess osc u ≤ ωn , Qn

(4.17)

4.4 The Recursive Argument

45

 2−p Proof. Recall the definition of a0 = 2ωλ and the construction of the initial cylinder so that the starting relation ess osc u ≤ ω

(4.18)

Q0

holds. We find  d

R 8

p

 ω 2−p Rp 2 8p   ω 2−p 2λ 2−p  ω 2−p Rp 1 = 2 ω1 2λ 8p  2−p  λ 2−p   2 ω ω1 2−p Rp = ω1 2 2λ 8p =

= σ p−2 2(λ−1)(2−p)−3p a1 Rp = a1 R1p , where R1 = C −1 R, provided C is chosen from C=σ

2−p p

2

(λ−1)(p−2) +3 p

>8.

From Proposition 4.6, we conclude ess osc u ≤ ess osc u ≤ σ ω = ω1 , p R Q1 Q(d( R 8 ) , 8 ) which puts us back to the setting of (4.18). The entire process can now be   repeated inductively starting from Q1 . Remark 4.8. The proof of Proposition 4.7 shows that it would have been sufficient to work with a number ω and a cylinder Q(a0 Rp , R) linked by (3.4). This relation is in general not verifiable a priori for a given cylinder, since its dimensions would have to be intrinsically defined in terms of the essential oscillation of u within it. The role of having introduced the cylinder Q(R2 , R) and having assumed (3.3) is that (3.4) holds true for the constructed box Q(a0 Rp , R). It is part of the proof of proposition 4.7 to show that, at each step, the cylinders Qn and the essential oscillation of u within them satisfy the intrinsic geometry dictated by (3.4). Lemma 4.9. There exist constants γ > 1 and α ∈ (0, 1), that can be determined a priori in terms of the data, such that for all the cylinders with 0 < ρ ≤ R , Q(a0 ρp , ρ) ,  ρ α u≤γω R . ess osc p Q(a0 ρ ,ρ)

46

4 Towards the H¨ older Continuity

Proof. Let 0 < ρ ≤ R be fixed. There exists a non-negative integer n such that C −(n+1) R ≤ ρ < C −n R ln σ , we deduce so, putting α = − ln C  ρ α n+1 ρ ρ ⇐⇒ σ α ≤ ⇐⇒ σ n+1 ≤ C −(n+1) ≤ . R R R Thus

 ρ α

, with γ = σ −1 . R To conclude the proof, observe that the cylinder Q(a0 ρp , ρ) is contained in the cylinder Qn = Q(an Rnp , Rn ), since ωn = σ n ω ≤ γ ω

ωn ≤ ω

ρ < C −n R = Rn .

and

  Let Γ = ∂p ΩT be the parabolic boundary of ΩT and u be a bounded local weak solution of (2.1) in ΩT , with M = u ∞,ΩT . Introduce the degenerate intrinsic parabolic p-distance from a compact set K ⊂ ΩT to Γ , by   p−2 1 p − dist(K; Γ ) := inf |x − y| + M p |t − s| p . (x,t)∈K (y,s)∈Γ

Theorem 4.10. Let u be a bounded local weak solution of (2.1) in ΩT and older continuous in ΩT , i.e., there exist M = u ∞,ΩT . Then u is locally H¨ constants γ > 1 and α ∈ (0, 1), depending only on the data, such that, for every compact subset K of ΩT , α  p−2 1 |x1 − x2 | + M p |t1 − t2 | p |u(x1 , t1 ) − u(x2 , t2 )| ≤ γM , p − dist(K; Γ ) for every pair of points (xi , ti ) ∈ K, i = 1, 2. Proof. Fix (xi , ti ) ∈ K, i = 1, 2, such that t2 > t1 and construct the cylinder   (x2 , t2 ) + Q M 2−p Rp , R . It is contained in ΩT if we choose R ≤ inf |x − y|

and

M

2−p p

x∈K y∈∂Ω

1

R ≤ inf t p . t∈K

Thus, in particular, we may choose 2R = p − dist (K; Γ ). To prove the H¨ older continuity in the t–variable assume first that t2 − t1 < M 2−p Rp .

4.5 Generalizations

47

Then, there exists ρ ∈ (0, R) such that t2 − t1 = M 2−p ρp , i.e., ρ=M

p−2 p

1

|t2 − t1 | p .

The oscillation inequality of Lemma 4.9, applied in the cylinder (x2 , t2 ) + Q(a0 ρp , ρ) implies

 |u(x2 , t2 ) − u(x2 , t1 )| ≤ γM

p−2

1

M p |t2 − t1 | p p − dist(K; Γ )

If (t2 − t1 ) ≥ M 2−p Rp , we have



|u(x2 , t2 ) − u(x2 , t1 )| ≤ 2M ≤ 4M

p−2

α .

1

M p |t2 − t1 | p p − dist(K; Γ )

 .

The H¨ older continuity in the space variables is proved analogously.

 

Remark 4.11. The theory includes statements of regularity up to the parabolic boundary of ΩT (cf. [14] and [52]).

4.5 Generalizations The analysis of the singular case 1 < p < 2 is somehow more involved, but several of the previous techniques apply (see [14] and [21]). As indicated earlier, the H¨ older continuity of u is solely a consequence of the energy inequalities (2.6) and the logarithmic inequalities (2.8). For this reason, the techniques just presented are rather flexible and adjust to a variety of singular and degenerate parabolic partial differential equations. A possible generalization is to equations with the full p−Laplacian type quasilinear structure ut − div a(x, t, u, ∇u) = b(x, t, u, ∇u)

in D (ΩT ),

(4.19)

where a : ΩT × Rd+1 → Rd and b : ΩT × Rd+1 → R are measurable and satisfy the structure assumptions (A1 ) (A2 ) (A3 )

a(x, t, u, ∇u) · ∇u ≥ C0 |∇u|p − ϕ0 (x, t); |a(x, t, u, ∇u)| ≤ C1 |∇u|p−1 + ϕ1 (x, t); |b(x, t, u, ∇u)| ≤ C2 |∇u|p + ϕ2 (x, t),

for a.e. (x, t) ∈ ΩT , with p > 1. The Ci , i = 0, 1, 2, are given positive constants and the ϕi , i = 0, 1, 2, are given non-negative functions, defined in ΩT and subject to the integrability conditions p

ϕ0 ,

ϕ1p−1 ,

ϕ2 ∈ Lq,r (ΩT )

48

4 Towards the H¨ older Continuity

with q, r ≥ 1 satisfying, for 1 < p ≤ d, d 1 + ∈ (0, 1). r pq See [14] for the details. Another family of equations to which the theory applies are degenerate or singular equations of porous medium type that can be cast in the form (4.19), for the structure assumptions (B1 ) (B2 ) (B3 )

a(x, t, u, ∇u) · ∇u ≥ C0 |u|m−1 |∇u|2 − ϕ0 (x, t), |a(x, t, u, ∇u)| ≤ C1 |u|m−1 |∇u| + ϕ1 (x, t); 2 |b(x, t, u, ∇u)| ≤ C2 |∇|u|m | + ϕ2 (x, t),

m > 0;

and the functions ϕi , i = 0, 1, 2, satisfy the same conditions as before with p = 2. We require     1,2 2 and |u|m ∈ L2loc 0, T ; Wloc (Ω) . u ∈ L∞ loc 0, T ; Lloc (Ω) There is a wide literature concerning this problem and we refer the reader to [3, 55] and the references therein. See also Chapter 6. Further generalizations can be obtained by replacing sm−1 , s > 0, with a function that blows up like a power when s  0 and is regular otherwise. To be specific, consider doubly degenerate equations of the form (4.19) with structure assumptions (C1 ) (C2 ) (C3 )

a(x, t, u, ∇u) · ∇u ≥ C0 Φ(|u|)|∇u|p − ϕ0 (x, t); 1 |a(x, t, u, ∇u)| ≤ C1 Φ(|u|)|∇u|p−1 + Φ p (u)ϕ1 (x, t); p |b(x, t, u, ∇u)| ≤ C2 Φ(|u|)|∇u| + ϕ2 (x, t).

Here ϕi , i = 0, 1, 2, satisfy the same conditions as before and the function Φ(·) is degenerate near the origin in the sense that ∃ σ > 0 : γ1 sβ1 ≤ Φ(s) ≤ γ2 sβ2 ,

∀ s ∈ (0, σ) ,

for given constants 0 < γ1 ≤ γ2 and 0 ≤ β2 ≤ β1 . For s > σ, i.e., away from zero, it is assumed that Φ is bounded above and below by given positive constants. We require that   1 and Φ p−1 (u)|∇u| ∈ Lploc (ΩT ) u ∈ Cloc 0, T ; L2loc (Ω) 1

and, denoting with F (·) the primitive of Φ p−1 (·), that   1,p (Ω) , F (u) ∈ Lploc 0, T ; Wloc which allows for an interpretation of the equation in the weak sense. One recognizes that if Φ(s) ≡ 1 the equation is of p-Laplacian type and if Φ(s) = older continuity sm−1 and p = 2 the equation is of porous medium type. The H¨ of solutions was obtained independently in [46] and [31].

5 Immiscible Fluids and Chemotaxis

This second part is devoted to a series of three applications of the method of intrinsic scaling to relevant models arising from flows in porous media, chemotaxis and phase transitions. We start with the flow of two immiscible fluids through a porous medium, proving the H¨ older continuity of the saturations, which satisfy a PDE with a two-sided degeneracy. The same type of structure arises in a model for the chemotactic movement of cells under a volume-filling effect and the extension to this case, which basically consists in dealing appropriately with an extra lower order term, is also included.

5.1 The Flow of Two Immiscible Fluids through a Porous Medium The flow of two immiscible fluids through a porous medium can be modeled through a parabolic equation with two degeneracies, namely vt − div (α(v)∇v) = 0,

(5.1)

where v ∈ [0, 1] and α(v) degenerates for v = 0 and v = 1. Following the important pioneering work in [35], that deals with the case of a strictly parabolic equation, a mathematical analysis of this model, where v represents the saturation of one of the fluids, was developed in [1]. The existence of a weak solution for the problem was established, together with the continuity of the saturation v under the assumption that α degenerates at most at one side, although no restrictions were put on the nature of the degeneracy. The result was extended to a setting where two degeneracies for α are allowed, provided some information on the nature of one of the degeneracies is assumed. First, in the same paper, the decay of α at one side was taken at most logarithmic and later, in [13], α was allowed to behave like a power. In [53] it was shown that v is locally H¨ older continuous if α decays

52

5 Immiscible Fluids and Chemotaxis

like a power at both degeneracies. We stress that the novelty lies in the H¨ older character of the solution since the continuity is well known. In fact, equation (5.1) can be written in the form
v with w= α(s) ds ; v = β(w), [β(w)]t − ∆w = 0 0

where β is an increasing function, for which the results of [22] apply. Some further physical motivation for the study of this type of equations, arising in polymer chemistry and combustion models, may be found in [30], where a numerical scheme is used to compute approximate solutions and interface curves for the Cauchy problem associated to (5.1) in a one dimensional case. From the mathematical point of view, the equation is interesting in that it exhibits a strong smoothing effect that prevents the possible development of singularities due to the degeneracies. The ultimate goal would be to obtain the continuity for a general quasi-linear equation, irrespective of the nature of the two degeneracies. This is also relevant in physical terms since the available experimental information about α is only of a qualitative nature, which makes all assumptions on the degeneracies quite restrictive (see [15]). We will consider here local weak solutions of equation (5.1), assuming that they exist; for the existence theory see [1]. Definition 5.1. A local weak solution of (5.1) is a measurable function v(x, t) defined in ΩT and such that   (i) v ∈ [0, 1] and v ∈ C 0, T ; L2 (Ω) ;
v   1 (ii) α(s) ds ∈ L2 0, T ; Hloc (Ω) ; 0

(iii) for every subset K ⊂ Ω and for every subinterval [t1 , t2 ] ⊂ (0, T ],
K

t2
vϕ dx + t1

t2

t1


{−v ϕt + α(v) ∇v · ∇ϕ} dx dt = 0, K

    1 0, T ; L2 (K) ∩ L2loc 0, T ; H01 (K) . for all ϕ ∈ Hloc We can write (iii) in an equivalent way that is technically more convenient and involves the discrete time derivative. Recalling definition (2.3) of the Steklov average of a function, the equivalent formulation reads (iii) for every compact K ⊂ Ω and for every 0 < t < T − h,
{(vh )t ϕ + (α(v)∇v)h · ∇ϕ} dx = 0, K×{t}

for all ϕ ∈ H01 (K).

(5.2)

5.2 Rescaled Cylinders

53

The regularity result will be obtained under the following assumptions on the diffusion coefficient α: (A1) α is a continuous function and α(v) > 0, for v ∈ (0, 1). (A2) ∃ δ0 ∈ (0, 12 ) such that, for constants 1 < C0 < C1 , C0 φ(v) ≤ α(v) ≤ C1 φ(v) ,

∀v ∈ [0, δ0 ]

and C0 ψ(1 − v) ≤ α(v) ≤ C1 ψ(1 − v) , (A3) ψ(s) = sp1 ;

φ(s) = sp2 ;

∀v ∈ [1 − δ0 , 1].

p1 > p2 > 0.

Although we are working with powers, the proof carries through with powerlike φ and ψ, i.e., functions that satisfy a condition of the type φ(v) ≥ C v φ (v). Our main result is stated next. Theorem 5.2. Under assumptions (A1)-(A3), any local weak solution of (5.1) is locally H¨ older continuous. The proof of the local H¨ older continuity uses the method of intrinsic scaling described in Part I. We will apply it here in a new setting since the nature of the degeneracies is quite different from the case of the p–Laplacian. For that reason, this chapter was kept essentially self-contained and can be read independently of most of Part I.

5.2 Rescaled Cylinders Consider a point (x0 , t0 ) ∈ ΩT and, by translation and to simplify, assume (x0 , t0 ) = (0, 0). Consider a small positive number > 0 and R > 0 such that Q((2R)2− , 2R) ⊂ ΩT and define µ− := ω :=

ess inf

Q((2R)2− ,2R)

ess osc

Q((2R)2− ,2R)

v;

µ+ :=

ess sup

Q((2R)2− ,2R)

v = µ+ − µ− .

v;

54

5 Immiscible Fluids and Chemotaxis

Construct the cylinder Q(θR2 , R),

with

θ−1 = φ

 ω  , 2m

where the number m will be chosen large, later in the proof, independently of ω. We assume µ+ = 1 and µ− = 0, since the other possibilities are clearly more favorable. We may assume that Q(θR2 , R) ⊂ Q((2R)2− , 2R), which means that −θR2 ≥ −(2R)2−

⇐⇒

θ−1 ≥ 2−2 R .

(5.3)

If this does not hold, then we have  ω  φ m < CR , 2 and then the oscillation would go to zero with R and there would be nothing to prove. We then have the relation ess osc v ≤ ω

(5.4)

Q(θR2 ,R)

which will be the starting point of the iteration process that leads to our main result. Note that we had to consider the cylinder Q((2R)2− , 2R) and assume (5.3), so that (5.4) would hold for the rescaled cylinder Q(θR2 , R). This is in general not true for a given cylinder since its dimensions would have to be intrinsically defined in terms of the essential oscillation of the function within it. Observe also that when the oscillation ω is small, and for m very large, then the cylinder Q(θR2 , R) is very long in the t direction. It is this feature that will allow us to accommodate the two degeneracies in the problem. We will also assume, without loss of generality, that ω < δ0 , where δ0 was introduced in (A2). We now consider subcylinders of Q(θR2 , R) of the form   2 ∗ R , R , with t∗ < 0. QtR := (0, t∗ ) + Q ψ( ω4 ) They are contained in Q(θR2 , R) if θR2 ≥ −t∗ + φ

R2 , ψ( ω 4)

which holds if

 ω  ω ≤ ψ( ) 2m 4

and t∗ is chosen such that t∗ ∈



 R2 R2 − , 0 . ψ( ω4 ) φ( 2ωm )

(5.5)

5.3 Focusing on One Degeneracy

We will assume further, and for technical reasons, that  ω  1 ω  φ m ≤ ψ 2 2 4

55

(5.6)

emphasizing that it is the choice of a big cylinder Q(θR2 , R), by choosing m very large, that makes (5.6) possible. The proof of the H¨ older continuity follows from the analysis of an alternative. We first concentrate on the degeneracy at 1 and later consider the behavior of v near 0. The common bottom line will be that going down to a smaller cylinder the oscillation decreases by a small factor that we can exhibit and that does not depend on the oscillation.

5.3 Focusing on One Degeneracy For a constant ν0 ∈ (0, 1), that will be determined depending only on the ∗ data, we will assume in this section that there is a cylinder of the type QtR for which ∗  ∗ ω  (5.7) ≤ ν0 QtR (x, t) ∈ QtR : v(x, t) > 1 − 2 leaving for the next section the analysis of the complementary case. We start by showing that if (5.7) holds then v is away from the degeneracy at 1 in a smaller cylinder of the same type. The next lemma specifies what this means. Lemma 5.3. There exists a constant ν0 ∈ (0, 1), depending only on the data, such that if (5.7) holds then v(x, t) < 1 −

ω , 4

∗

a.e. in QtR . 2

Proof. Let vω := min{v, 1− ω4 }. Take the cylinder for which (5.7) holds, define Rn =

R R + n+1 , 2 2

n = 0, 1, . . . ,

and construct the family of nested and shrinking cylinders   Rn2 t∗ ∗ ∗ ,t . QRn = KRn × t − ψ( ω4 ) Consider piecewise smooth cutoff functions 0 ≤ ξn ≤ 1, defined in these cylinders, and satisfying the following set of assumptions ∗

∗

ξn = 1 in QtRn+1 ; |∇ξn | ≤ 22(n+1) |∆ξn | ≤ ; R2

2n+1 ; R

ξn = 0 on ∂p QtRn ; (5.8) ψ( ω ) 0 ≤ (ξn )t ≤ 22(n+1) 42 . R

56

5 Immiscible Fluids and Chemotaxis

Let

ω ω − n+2 , 4 2

kn = 1 −

n = 0, 1, . . .

choose as test function in (5.2) ϕ = [(vω )h − kn ]+ ξn2 and integrate in time over (t∗ −

2 Rn , t) ψ( ω 4)

for t ∈ (t∗ −

2 Rn , t∗ ). ψ( ω 4)

τn := t∗ −

Putting Rn2 ψ( ω4 )

the first term gives (for K = KRn and omitting dx and dt throughout)


t


(vh )t [(vω )h −

τn

KRn

kn ]+ ξn2

1 = 2




t

τn


KRn



2

[(vω )h − kn ]+

 t

ξn2

   
t
 ω  ω v − (1 − ) + 1 − − kn ξ2 . 4 4 + h t n τn KRn Next, integrate by parts and let h → 0. Using the properties of the Steklov average, we get

1 1 (vω − kn )2+ ξn2 − (vω − kn )2+ ξn2 2 KRn ×{t} 2 KRn ×{τn }
t
2 − (vω − kn )+ ξn (ξn )t τn

KRn

 
  ω  ω v− 1− + 1 − − kn ξ2 4 4 + n KRn ×{t}
  ω  v− 1− − ξ2 4 + n KRn ×{τn } 
t
  ω  v− 1− −2 ξn (ξn )t 4 + τn KRn 

Since the second and the fifth terms vanish, due to the fact that ξn was ∗ chosen such that it vanishes on the parabolic boundary of QtRn , and the fourth term is positive, we estimate from below, using the other assumptions on ξn , by
 ω  22(n+1)  ω 2
t
1 (vω − kn )2+ ξn2 − ψ χ{vω ≥kn } 2 KRn ×{t} 4 R2 4 τn KRn  ω  22(n+1)  ω 2
t
−2ψ χ{v≥1− ω4 } = (∗). 4 R2 4 τn KRn

5.3 Focusing on One Degeneracy

Observe that ω , 4

(vω − kn )+ ≤

1−

Finally, remarking that v ≥ 1 − (∗) ≥

1 2

and

v − (1 −

ω  ω ) ≤ . 4 + 4

⇒ vω ≥ kn , we obtain

ω 4


(vω − kn )2+ ξn2 − 3ψ

KRn ×{t}



ω ω − kn ≤ 4 4

57

 ω  22(n+1)  ω 2
t
χ{vω ≥kn } . 4 R2 4 τn KRn

Concerning the diffusion term, we first pass to the limit in h, obtaining
t


(α(v)∇v)h · ∇ [(vω )h − kn ]+ ξn2


t

τn






α(v)∇v · ξn2 ∇(vω − kn )+ + 2(vω − kn )+ ξn ∇ξn

−→


t




τn

KRn

KRn




t

2




α(v) |ξn ∇(vω − kn )+ | + 2

= τn

α(v)(vω−kn )+ ξn ∇v·∇ξn . τn

KRn

KRn

Next, we estimate the second term:

t α(v)(vω − kn )+ ξn ∇v · ∇ξn 2 τn KRn


t


|α(v)| (vω − kn )+ ξn |∇ξn | |∇(vω − kn )+ |

≤2 τn

KRn


   v 
t
ω · ∇ξn + 2 1 − − kn ξn ∇ α(s) ds 4 τn KRn 1− ω 4 + 



t ω 1 t 2 2 2 |ξn ∇(vω − kn )+ | + (vω − kn )+ |∇ξn | ≤ C1 ψ 2 τn KRn τn KRn 
 v ω
t
  |∇ξn |2 + ξn ∆ξn +2 α(s) ds − ω 4 τn KRn 1− 4

≤ C1 ψ

ω 
2

t

+




|ξn ∇(vω − kn )+ | τn

C1 22(n+1) ψ( ω2 ) + R2

KRn

 ω 2
4

t

τn

2


χ{vω ≥kn }

KRn

 ω  22(n+1)  ω   ω 
t
2 +2 ψ χ{vω ≥kn } , 4 R2 4 4 τn KRn

58

5 Immiscible Fluids and Chemotaxis

since






v

α(s) ds 1− ω 4

+

 ω  ω  ≤ ψ 1 − (1 − ) 1 − (1 − ) . 4 4

Observing that ∇(vω − kn )+ is only nonzero in the set {kn < v < 1 − ω4 }, and that in this set  ω ω  α(v) ≥ C0 ψ(1 − v) ≥ C0 ψ 1 − (1 − ) = C0 ψ , 4 4 we conclude, choosing =

(C0 − 12 )ψ( ω4 ) , C1 ψ( ω2 )

that


t

τn






 1 ω  t  2 2 α(v) ∇v · ∇ (vω − kn )+ ξn ≥ ψ |ξn ∇(vω − kn )+ | 2 4 τn KRn KRn    

ω 2     2(n+1) C1 ψ( 2 ) 2 ω 2 t ω − χ{vω ≥kn } . + 4ψ 4 R2 4 (C0 − 12 )ψ( ω4 ) τn KRn

Now, putting both estimates together, we arrive at
ess sup

τn ≤t≤t∗

KRn ×{t}

(vω − kn )2+ ξn2 + ψ

ω
4

t∗


2

|ξn ∇(vω − kn )+ | τn

KRn

   2  ω  22(n+1)  ω 2
t∗
C1 ψ( ω2 ) ≤2 χ{vω ≥kn } . + 7ψ 4 R2 4 (C0 − 12 )ψ( ω4 ) τn KRn Next we perform a change in the time variable, putting t = (t − t∗ )ψ( ω4 ) and define vω (·, t) = vω (·, t) and ξn (·, t) = ξn (·, t), obtaining the simplified inequality   (vω − kn )+ ξn 2 2 2 ,R )) V (Q(Rn n

  ω 2 2     ψ( 2 ) C1 22(n+1) ω 2 0 ≤ 2 (C − 1 ) ψ( ω ) + 7 χ{vω ≥kn } . R2 4 −R2 KR 0

2

n

4

Define, for each n,
An =

0

2 −Rn


KRn

χ{vω ≥kn } dx dt

n

(5.9)

5.3 Focusing on One Degeneracy

and observe that the following estimates hold  ω 2 1 2 An+1 = |kn+1 − kn | An+1 2(n+2) 4 2 2 ≤ (vω − kn )+ 2,Q(R2 ,Rn+1 ) n+1 2    ≤ (vω − kn )+ ξn 2,Q(R2 ,Rn ) n 2 2  ≤ C (vω − kn )+ ξn V 2 (Q(R2 ,R )) And+2 n n    ω 2 2 2(n+1)  2 ψ( ) C1 2 ω 1+ 2 2 ≤ 2C +7 × An d+2 . ω 1 2 R 4 (C0 − 2 ) ψ( 4 )

59

(5.10)

In fact, the second inequality is obvious; the first one holds due to the fact that kn < kn+1 ; the third inequality is a consequence of Theorem 2.11, and the last one follows from (5.9). Next, define the numbers Xn =

An , |Q(Rn2 , Rn )|

2 divide (5.10) by Q(Rn+1 , Rn+1 ) and obtain the recursive relation 2 1+ d+2

Xn+1 ≤ γ 42n Xn 

where γ=C

,

  ω 2 ψ( 2 ) C12 +7 . (C0 − 12 ) ψ( ω4 )

We can use Lemma 2.9 to conclude that if X0 ≤ γ −

d+2 2

4−2(

d+2 2 2 )

=: ν0

(5.11)

then Xn −→ 0.

(5.12)

But (5.11) is nothing but the assumption (5.7) of the lemma and the conclusion easily follows from (5.12). In fact, observe that Rn 

R 2

and

kn  1 −

ω , 4

and since (5.12) implies that An → 0, we conclude that      2 R R ω : vω (x, t) ≥ 1 − , (x, t) ∈ Q 2 2 4  ∗ ω  = (x, t) ∈ QtR : v(x, t) ≥ 1 − =0 2 4 and the lemma is proved.

60

5 Immiscible Fluids and Chemotaxis

Let us just show why ν0 is independent of ω, since this is crucially related to the fact that ψ is a power. In fact, we have  ν0 = γ

− d+2 2

2 −2( d+2 2 )

4

2 −2( d+2 2 )

=4

C

− d+2  ω 2 2 ψ( 2 ) C12 + 7 , (C0 − 12 ) ψ( ω4 )

and so we conclude showing that ψ( ω2 )  ω p1 = ψ( ω4 ) 2

 p1 4 = 2p1 ω  

is independent of ω.

Our next aim is to show that the conclusion of Lemma 5.3 holds in a full cylinder of the type Q(τ, ρ). The idea is to use the fact that at the time level − t := t∗ −

 R 2 2   ψ ω4

(5.13)

the function v(x) is strictly below the level 1 − ω4 in the cube K R and look at 2 this time level as an initial condition to make the conclusion hold up to t = 0, eventually shrinking the cube. Again this is a sophisticated way of showing that somehow the equation behaves like the heat equation. As an intermediate step we need the following lemma. Lemma 5.4. Given ν1 ∈ (0, 1), there exists s1 ∈ N, depending only on the data, such that  ω  t, 0). x ∈ K R4 : v(x, t) > 1 − s1 ≤ ν1 K R4 , ∀ t ∈ (− 2 Proof. We use a logarithmic estimate for the function (v − k)+ in the cylinder Q( t, R2 ), with the choices k =1−

ω 4

and

c=

ω , 2n+2

where n ∈ N will be chosen later, as parameters in the standard logarithmic function (2.7). We have  ω  ω + (5.14) = ess sup v − 1 + ≤ . v − k ≤ Hv,k 4 + 4 Q( t, R ) 2

+ + If Hv,k ≤ ω8 , the result is trivial for the choice s1 = 3. Assuming Hv,k > ω8 , recall from section 2.3 that the logarithmic function ψ + (v) is defined in the + − v + k + c > 0), and whole domain of v, Q( t, R2 ) (since it is obvious that Hv,k given by

5.3 Focusing on One Degeneracy

ψ +H + ,k, { v,k

⎧  ⎪ ⎪ ⎪ ⎨ ln ω 2n+2

}

(v) =

⎪ ⎪ ⎪ ⎩

61



+ Hv,k + Hv,k −v+k+

ω

if v > k +

ω 2n+2

if v ≤ k +

ω 2n+2 .

2n+2

0

From (5.14), we can easily estimate ψ (v) ≤ n ln 2 +

+ Hv,k

since

+ Hv,k

−v+k+

ω 2n+2

≤

ω 4 ω

= 2n

2n+2

and the derivative (here in the nonvanishing case v > k + c)   1 2n+2 −1 +  = . (v) = + ψ ≤ Hv,k − v + k + c c ω To obtain the estimate, choose a cutoff function 0 < ζ(x) ≤ 1, defined on K R and such that 2

and

ζ = 1 in K R 4

|∇ζ| ≤

C R,



t, t), and multiply (5.2) by 2ψ + (vh ) (ψ + ) (vh ) ζ 2 . Integrating in time in (− with t ∈ (− t, 0), and performing the usual integrations by parts and passages to the limit in h for the Steklov averages, we obtain, concerning the time part,

 + 2 2  + 2 2 ψ (v) ζ − ψ (v) ζ . K R ×{− t}

K R ×{t} 2

2

Now observe that as a consequence of Lemma 5.3, we have v(x, − t) < k in the cube K R , which implies that 2

 +  ψ (v) (x, − t) = 0, x ∈ K R . 2

As for the space part, we start by passing to the limit in h, thus getting
t
    α(v)∇v · ∇ 2ψ + (v) ψ + (v) ζ 2
=

− t KR

t




2

− t KR




+2

   2  α(v)|∇v|2 2 1 + ψ + (v) (ψ + ) (v) ζ 2




2

t

− t KR 2

    α(v)∇v · ∇ζ 2ψ + (v) ψ + (v) ζ = (∗).

62

5 Immiscible Fluids and Chemotaxis

Using Young’s inequality, we estimate the second term as
t
   +  + 2 α(v)∇v · ∇ζ 2ψ (v) ψ (v) ζ −t K R 2


≤2

t




 2 α(v)|∇v| ζ ψ (v) (ψ + ) (v) + 2 2 2

− t KR




2

− t KR 2

C R2




− t KR

α(v)|∇ζ|2 ψ + (v),

2

and as a consequence we get
t

 2 (∗) ≥ 2 α(v)|∇v|2 ζ 2 (ψ + ) (v) − 2

≥ −2n ln 2

t

+




t




− t KR

α(v)|∇ζ|2 ψ + (v)

2

t




− t KR

α(v)χ{v>1− ω4 } ≥ −2n ln 2

ω  C  , t K R C1 ψ 2 2 R 4

2

because, in the relevant set,   ω  ω  α(v) ≤ C1 ψ(1 − v) ≤ C1 ψ 1 − 1 − = C1 ψ . 4 4 Now we observe that, due to our choice of t∗ ,  R 2 ∗  t = −t + 2 ω  ≤ θR2 , ψ 4 and so we can condensate the two estimates in the logarithmic estimate
ω   + 2 2 ψ (v) ζ ≤ C n θ K R ψ . (5.15) sup 2 4 K R ×{t} − t≤t≤0 2

Now, since the integrand is nonnegative, we estimate below the left hand side of (5.15) integrating over the smaller set  ω  S = x ∈ K R : v(x, t) > 1 − n+2 ⊂ K R . 4 2 2 Observing that in S, ζ = 1 and  + 2   n−1 2 ψ (v) ≥ ln 2 = (n − 1)2 (ln 2)2 , we get

ω  (n − 1)2 (ln 2)2 |S| ≤ C n θ K R ψ 2 4

and consequently,  x ∈ K R4 : v(x, t) > 1 −

  ψ ω4 n ω    R . ≤ C K 4 2n+2 (n − 1)2 φ 2ωm

5.3 Focusing on One Degeneracy

63

To prove the lemma we just need to choose s1 = n + 2 since if n ≥ 1 +

2 α

  2C ψ ω4 ,  n≥1+ ν1 φ 2ωm

with

then n ≤ α, (n − 1)2

α > 0.

Concerning the independence of ω, observe that ω  ω p1 ψ 4  ω  =  ω4 p2 = 2mp2 −p1 ω p1 −p2 ≤ 2mp2 −p1 , φ m 2 2m because p1 > p2 and ω < 1, and this expression does not depended on ω. We strongly emphasize that it is crucial at this point that both degeneracies are   powers and that p1 > p2 . We now arrive at the main result of this section that establishes the first alternative. Proposition 5.5. There exists a constant s1 ∈ N, depending only on the data, such that if (5.7) holds then   R ω  . v(x, t) < 1 − s1 +1 , a.e. in Q t, 2 8 Proof. Consider the cylinder for which (5.7) holds, let Rn =

R R + n+3 , 8 2

n = 0, 1, . . .

t and construct the family of nested and shrinking cylinders Q( t, Rn ), where  is given by (5.13). Take piecewise smooth cutoff functions 0 < ζn (x) ≤ 1, not depending on t, defined in KRn and satisfying the assumptions ζn = 1 in KRn+1 , |∇ζn | ≤ Take also

2n+4 R

and

|∆ζn | ≤

22(n+4) R2 .

ω ω − s1 +1+n , n = 0, 1, . . . , 2s1 +1 2 (where s1 > 1 is to be chosen) and derive local energy inequalities similar to those obtained in Lemma 5.3, now for the functions  ω  with vω := min v, 1 − s1 (vω − kn )+ ζn2 , 2 kn = 1 −

64

5 Immiscible Fluids and Chemotaxis

  and in the cylinders Q  t, Rn . Observing that, due to Lemma 5.3, we have ω v(x, − t) < 1 − ≤ kn 4

in the cube K R ⊃ KRn , 2

which implies that t) = 0, x ∈ KRn , (v − kn )+ (x, −

n = 0, 1, . . . .

Performing the same type of reasoning used in Lemma 5.3 (which we shall not repeat here), we get
sup

− t k = ω2 } and using (5.21). Now, since the integrand is nonnegative, we estimate below the left hand side in the expression above integrating over the smaller set  ω  S = x ∈ K(1−σ)R : v(x, t) < n+1 ⊂ KR , t ∈ (t◦ , t∗ ). 2 Observing that in S, ζ = 1 and  − 2   n−1 2 ψ (v) ≥ ln 2 = (n − 1)2 (ln 2)2 , we get   n 1 φ ω2   |KR | |KR | + C |S| ≤ C (n − 1)2 σ 2 ψ ω4   2    1 − ν0 1 1 φ ω2 n   |KR | . |KR | + C ≤C n−1 1 − ν20 n σ 2 ψ ω4 

n n−1

2 

1 − ν0 1 − ν20



On the other hand, we have  ω  x ∈ KR : v(x, t) < n+1 2  ≤ x ∈ K(1−σ)R : v(x, t) < ≤ |S| + dσ|KR |,

ω  + KR \ K(1−σ)R n+1 2

so the conclusion is that for all t ∈ (t◦ , t∗ ),  x ∈ KR : v(x, t) <  ≤C

n n−1

2 

1 − ν0 1 − ν20



ω 

2n+1

   1 1 φ ω2   + dσ |KR | . + n σ 2 ψ ω4

5.4 Behaviour Near the other Degeneracy

71

Now, choose σ so small that Cdσ ≤ 38 ν02 and n so large that  C

n n−1

2

ν0  (1 + ν0 ) =: β > 1 ≤ 1− 2 

and

  C φ ω2 3   ≤ ν02 nσ 2 ψ ω4 8

and the lemma follows with s2 = n + 1. Concerning the independence of s2 with respect to ω, and since ν0 has already been chosen respecting that independence, it is enough to note that, due to (5.6),     φ ω2 φ ω2   ≤  ω  = 2(m−1)p2 , ψ ω4 φ 2m expression does not depended on ω.

 

Now we want to show that the same type of conclusion  holds in an upper portion of the full cylinder Q(θR2 , R), say for all t ∈ − θ2 R2 , 0 . We just ∗ have to use the fact that (5.20) holds for all cylinders of the type QtR and so, recalling (5.5), the conclusion of the previous lemma holds true for all time levels R2 ν0 R 2 R2 − − . t≥ ψ( ω4 ) φ( 2ωm ) 2 ψ( ω4 ) Using assumption (5.6), we have

  1 ν0  R2 1− −1 2 2 φ( 2ωm )  ν0  R2 = − 1+ 2 2φ( 2ωm ) θ ≤ − R2 . 2   Corollary 5.9. For all t ∈ − θ2 R2 , 0 ,    ν 2  ω  0 |KR | . x ∈ KR : v(x, t) < s2 ≤ 1 − 2 2 R2 ν0 R 2 R2 − ≤ ω − ω ψ( 4 ) φ( 2m ) 2 ψ( ω4 )

The previous result means that the set of degeneracy at 0 does not fill the entire cube KR , for all t ∈ − θ2 R2 , 0 . The next result uses this stability information. Proposition 5.10. For every λ0 ∈ (0, 1), there exist constants s2 < s3 ∈ N and m0 ∈ N, depending only on the data, such that, if m > m0 then      (x, t) ∈ Q θ R2 , R : v(x, t) < ω ≤ λ0 Q θ R2 , R . 2 2s3 2

72

5 Immiscible Fluids and Chemotaxis

Instead of proving this proposition directly for v we will formulate an equivalent result for a new unknown function defined by
v α(s) ds, v ∈ [0, δ0 ]. w = A(v) := 0

We find that w satisfies the equation [B(w)]t − ∆w = 0,

in D (ΩT ),

(5.23)

where B = A−1 is the inverse of A; both functions are monotone increasing. Define also
ω ω := α(s) ds, 0

and rephrase Corollary 5.9 in terms of w as follows. Corollary 5.11. There  exists a number s4 ∈ N, depending only on the data, such that, for all t ∈ − θ2 R2 , 0 ,    2  x ∈ KR : w(x, t) < ω ≤ 1 − ν0 |KR | . 2s4 2 This fact is a consequence of the following simple reasoning:
ωs2
ωs2 2 2 ω α(s) ds = sp2 ds v < s2 ⇐⇒ w < 2 0 0
ω  ω p2 +1 α(s) ds s2 = 0 s (p +1) = 2 p2 + 1 22 2 ω = s4 , with s4 = s2 (p2 + 1). 2 Proposition 5.12. For every λ0 ∈ (0, 1), there exist constants s5 ∈ N and m0 ∈ N, depending only on the data, such that, if m > m0 then      (x, t) ∈ Q θ R2 , R : w(x, t) < ω ≤ λ0 Q θ R2 , R . s 2 25 2 Proof. We derive an energy inequality from equation (5.23) and use again the iteration technique. To be entirely precise, we should argue with the Steklov averages, as before, but to simplify we only do the formal computations this time. Consider a piecewise smooth cutoff function 0 ≤ ζ ≤ 1, defined in Q(θR2 , 2R), and satisfying the following set of assumptions ζ = 1 in Q( θ2 R2 , R); |∇ζ| ≤

1 ; R

ζ = 0 on ∂p Q(θR2 , 2R); 0 ≤ ζt ≤ C

φ( 2ωm ) . R2

5.4 Behaviour Near the other Degeneracy

73

Let

ω , (l > s4 to be chosen), 2l multiply (5.23) by −(w − k)− ζ 2 and integrate in time over (−θR2 , 0). Concerning the time part, we get (formally)
0
  [B(w)]t −(w − k)− ζ 2 −θR2 K2R 


k=

0

(w−k)−

= −θR2

K2R







0

B  (k − s)s ds







−

K2R ×{−θR2 }


−2




0 −θR2

ω ≥ −C l B 2



ω 2l




K2R



ζ2



(w−k)−

= K2R ×{0}

B  (k − s)s ds

0

t

ζ2 

(w−k)−



B (k − s)s ds

0 (w−k)−

ζ2

 B  (k − s)s ds

ζζt

0

φ( 2ωm ) R2

  Q θ R2 , R , 2

as a consequence of the following three facts: ζ(x, −θR2 ) = 0, due to our choice of ζ,
(w−k)− B  (k − s)s ds ≥ 0, 0

since B  ≥ 0, and
(w−k)−
 B (k − s)s ds ≤ (w − k)− 0

(w−k)−

B  (k − s) ds

0

= (w − k)− [−B (k − (w − k)− ) + B(k)] = (w − k)− [B(k) − B(w)]   ω ω ≤ kB(k) = l B . 2 2l From the space part, comes

0
  ∇w · ∇ −(w − k)− ζ 2 = −θR2

K2R


−2

1 ≥ 2




0

−θR2




0 −θR2

2

−θR2




|ζ ∇(w − k)− | K2R

(w − k)− ζ ∇w · ∇ζ K2R

 2

|ζ ∇(w − k)− | − C K2R




0

ω 2l

2

1 R2

  Q θ R2 , R , 2

74

5 Immiscible Fluids and Chemotaxis

and we combine both estimates in
0
−θR2

|ζ ∇(w − k)− |

2

K2R

    θ 2 ω ω φ( 2ωm ) R ≤ C lB , R Q l 2 2 2 R 2    2 θ 2 ω 1 . R , R Q +C l 2 2 R 2   Integrating over the smaller set Q θ2 R2 , R , where ζ = 1, we arrive at 2

∇(w − k)− 2,Q( θ R2 ,R) 2      2

l C 2 ω ω ω φ ≤ 1+ B l m 2 ω 2 2 R 2l

  Q θ R2 , R . 2

(5.24)

Let us now use Lemma 2.7, with the function w(x, t) defined for t ∈ [− θ2 R2 , 0], and the levels k1 =

ω 2l+1

and

k2 = k =

ω ; l = s4 , s4 + 1, . . . 2l

We know from Corollary 5.11, and since l ≥ s4 , that   x ∈ KR : w(x, t) > ω ≥ |KR | − x ∈ KR : w(x, t) < ω l s 2 24    ν 2 0 |KR | ≥ |KR | − 1 − 2  ν 2 0 |KR | , = 2 for all t ∈ [− θ2 R2 , 0]. Next, define

Al (t) =

x ∈ KR : w(x, t)


ω , 2s3 +1

a.e. in QjR0 . 2

5.4 Behaviour Near the other Degeneracy

77

Proof. Let vω := max{v, 2s3ω+2 }. Define Rn =

R R + n+1 , 2 2

n = 0, 1, . . . ,

and construct the family of nested and shrinking cylinders   Rn2 Rn2 j0 , −(j0 − 1) . QRn = KRn × −j0 µ µ Consider piecewise smooth cutoff functions 0 ≤ ζn ≤ 1, defined in these cylinders, and satisfying the following set of assumptions ζn = 1 in QjR0n+1 ; |∇ζn | ≤ |∆ζn | ≤

22(n+1) ; R2

Let kn =

ζn = 0 on ∂p QjR0n ;

2n+1 ; R

0 ≤ (ζn )t ≤ 22(n+1) ω

2s3 +1

+

ω 2s3 +1+n

,

µ . R2

n = 0, 1, . . .

choose as test function in (5.2) ϕ = − [(vω )h − kn ]− ζn2 and proceed as in the proof of Lemma 5.3 to get 2  ω 
−(j0 −1) Rµn
2 ess sup (vω −kn )2− ζn2 +φ s3 +2 |ζn ∇(vω − kn )− | 2 Rn 2 KRn ×{t} −j0 µ KRn




2 Rn

22(n+1)  ω 2  ω  −(j0 −1) µ ≤C φ χ{vω ≤kn } . R2 R2 2s3 +2 2s3 −j0 µn KRn   R2 Now perform a change in the time variable, putting t = t + (j0 − 1) µn µ and define vω (·, t) = vω (·, t) and ζn (·, t) = ζn (·, t),

and obtain the inequality, with Γ = Γ

d+2 2

φ1 φ2 ,


KRn ×{t}

≤C




2

ess sup

(vω − kn )2− ζn +

22(n+1)  ω 2 Γ R2 2s3




0

2 −Rn

0




2 −Rn

ζn ∇(vω − kn )− 2

KRn


KRn

χ{vω ≤kn } .

The conclusion of the proof follows from a refinement of the iteration technique used in the previous results; it can be found in [13].  

78

5 Immiscible Fluids and Chemotaxis

Remark 5.14. Observe that (5.27) plays the same role as the assumption in the first alternative and that Lemma 5.13 is the corresponding analogue of Lemma 5.3. The next two results, leading to the conclusion of the second alternative, reproduce the ideas in Lemma 5.4 and Proposition 5.5, i.e., we first use the logarithmic estimates to extend the result to a full cylinder in time and then, with the aid of the energy estimates, conclude that v is strictly away from 0 in a cylinder of the type Q(τ, ρ). Lemma 5.15. Given ν1 ∈ (0, 1), there exists s6 ∈ N, depending only on the data, such that   R 2   ω  x ∈ K R4 : v(x, t) ≤ s6 ≤ ν1 K R4 , ∀t ∈ −j0 2 , 0 . 2 µ Proof. To simplify, put

 R 2 t˜ = j0

2

µ

.

We use a logarithmic estimate for the function (v−k)− in the cylinder Q(t˜, R2 ), with the choices ω ω and c = n+1 , k = s3 +1 2 2 where n ∈ N will be chosen later, as parameters in the standard logarithmic function(see (2.7)). Choose a piecewise smooth cutoff function 0 < ζ(x) ≤ 1, defined on K R 2 and such that ζ = 1 in K R and |∇ζ| ≤ C R, 4



and multiply (5.2) by 2ψ − (vh ) (ψ − ) (vh ) ζ 2 . As in the proof of with the obvious changes, we get
 ω  t˜  − 2 2 ψ (v) ζ ≤ C (n − s3 ) φ s3 +1 sup 2 R2 K R ×{t} −t˜≤t≤0

Lemma 5.4, K R2

2

  φ 2s3ω+1  ω  K R ≤ C (n − s3 ) 2 φ 2m since, recalling (5.26),  R 2 t˜ = j0

2

µ

n0 ≤ µ



R 2

2

 R 2 = 2 ω  . φ 2m

Now, since the integrand is nonnegative, we estimate below the left hand side of the inequality integrating over the smaller set  ω  S = x ∈ K R : v(x, t) ≤ n+1 ⊂ K R 4 2 2

5.4 Behaviour Near the other Degeneracy

79

and, observing that in S, ζ = 1 and  − 2 ψ (v) ≥ (n − s3 )2 (ln 2)2 , we get  x ∈ K R4 : v(x, t) ≤

  φ 2s3ω+1 1 ω    R . ≤ C K 4 2n+1 n − s3 φ 2ωm

To prove the lemma choose s6 = n + 1

n ≥ s3 +

with

  C φ 2s3ω+1 ω  . ν 1 φ 2m

Concerning the independence of ω, observe that s3 was chosen independently of ω and  ω p2  ω  φ s3 +1 s3 +1 2 ω  = 2 ω p2 = 2(m−s3 −1)p2 , φ m 2 2m  

expression that does not depended on ω.

Proposition 5.16. There exists a constant s6 ∈ N, depending only on the data, such that   ω R . v(x, t) > s6 , a.e. in Q t˜, 2 8 Proof. Define Rn =

R R + n+3 , 8 2

n = 0, 1, . . .

and construct the family of nested and shrinking cylinders Q(t˜, Rn ). Take piecewise smooth cutoff functions 0 < ζn (x) ≤ 1, independent of t, defined in KRn and satisfying the assumptions ζn = 1 in KRn+1 , |∇ζn | ≤ Take also

2n+4 R

and

|∆ζn | ≤

22(n+4) R2 .

ω ω + s6 +1+n , n = 0, 1, . . . , 2s6 +1 2 (where s6 > 1 is to be chosen) and derive local energy inequalities similar to those obtained in Proposition 5.5, now for the functions  ω  with vω := max v, s6 +1 −(vω − kn )− ζn2 , 2 kn =

80

5 Immiscible Fluids and Chemotaxis

  and in the cylinders Q t˜, Rn . Using the same reasoning, we get
sup −t˜ 1. u ∈ Lq 0, T ; W 2,q (Ω) , Putting these estimate together with the ones obtained in the previous section, we arrive at
 ω 
t∗
2 (vω − kn )2+ ξn2 + ψ |ξn ∇(vω − kn )+ | ess sup ∗ 4 τn ≤t≤t KRn ×{t} τn KRn    2  ω  22(n+1)  ω 2
t∗
C1 ψ( ω2 ) + (7 + M )ψ ≤2 χ{vω ≥kn } (C0 − 1)ψ( ω4 ) 4 R2 4 τn KRn M 2 + 2M   ∇u 2Lp (ΩT ) + ψ ω4




t∗

τn

+ Akn ,Rn (σ) dσ

1− p2 .

Next we perform a change in the time variable, putting t = (t − t∗ )ψ( ω4 ), and define vω (·, t) = vω (·, t) and ξn (·, t) = ξn (·, t), to obtain the simplified inequality   (vω − kn )+ ξn 2 2 2 ,R )) V (Q(Rn n    2 ψ( ω2 ) C12 22(n+1)  ω 2 ≤2 +7+M An ω (C0 − 1) ψ( 4 ) R2 4   ω  p2 −2 1− 2 +(M 2 + 2M ) ∇u 2Lp (ΩT ) ψ An p , 4 defining, for each n,
0
χ{vω ≥kn } dx dt. An = 2 −Rn

KRn

Next, observe that the following estimates hold

5.5 A Problem in Chemotaxis

1

 ω 2

22(n+2)

4

2

An+1 = |kn+1 − kn | An+1

85

(5.30)

2

≤ (vω − kn )+ 2,Q(R2

n+1 ,Rn+1 )

2  ≤ (vω − kn )+ ξn 2,Q(R2 ,Rn ) n

2 2  ≤ C (vω − kn )+ ξn V 2 (Q(R2 ,Rn )) And+2 n    ω 2 2 2 ψ( 2 ) C1 22(n+1)  ω 2 1+ d+2 ≤ 2C + 7 + M An ω 2 (C0 − 1) ψ( 4 ) R 4

  ω  p2 −2 1− 2 + 2 +C(M 2 + 2M ) ∇u 2Lp (ΩT ) ψ An p d+2 . 4 The reasoning is the same as before. Next, define the numbers 1/p

An An ; Zn = , 2 |Q(Rn , Rn )| |KRn | 2 divide (5.30) by Q(Rn+1 , Rn+1 ) and obtain the recursive relation

 2 1+ 2 Xn+1 ≤ γ 42n Xn d+2 + Xnd+2 Zn1+κ , n = 0, 1, 2, . . . Xn =

where κ = p − 3 > 0 and   ω 2 ψ( 2 ) C12 +7+M ; γ = C max (C0 − 1) ψ( ω4 )

 ω −2   ω  p2 −2 ψ Rdκ 4 4

 .

A similar reasoning leads to

 Zn+1 ≤ γ 42n Xn + Zn1+κ ,

n = 0, 1, 2, . . .

We can now use Lemma 2.10 to conclude that if X0 + Z01+κ ≤ (2γ)−

1+κ θ

4

−2(1+κ) θ2

=: ν0 ,

θ = min

 2 ;κ d+2

(5.31)

then Xn , Zn −→ 0.

(5.32)

But (5.31) follows from the assumption in the first alternative and the conclusion is a direct consequence of (5.32). It remains to show that ν0 , i.e., γ, is independent of ω, which is crucially related to the fact that ψ is a power. In fact, we have  p1 ψ( ω2 )  ω p1 4 = = 2p1 . ω ψ( 4 ) 2 ω

86

5 Immiscible Fluids and Chemotaxis

On the other hand, we can assume, without loss of generality, that  ω −2   ω  p2 −2 ψ Rdκ ≤ 1. 4 4 Otherwise, we would have ω < CRα , with α = result would be trivial.

dκp 2p+2pp1 −2p1

> 0, and the

6 Flows in Porous Media: The Variable Exponent Case

Partial differential equations with nonlinearities involving variable exponents have attracted an increasing amount of attention in recent years. The development, mainly by R˚ uˇziˇcka [47], of a theory modeling the behavior of electrorheological fluids, an important class of non-Newtonian fluids, seems to have boosted a still far from completed effort to study and understand this type of equations. Other important applications relate to image processing [8], elasticity [56] or flows in porous media [2]. We will consider the parabolic equation in divergence form   (6.1) ut − div |u|γ(x,t) ∇u = 0, with a variable exponent of nonlinearity γ, which is a generalization of the famous porous medium equation and occurs as a model for the flow of an ideal barotropic gas through a porous medium. The main feature in equation (6.1) is that it is degenerate due to the exponential nonlinearity: the diffusion coefficient |u|γ(x,t) vanishes at points where u = 0. Results on the existence and uniqueness of weak solutions of (6.1), together with some localization properties, were obtained by Antontsev and Shmarev [2]. Under appropriate assumptions, we use intrinsic scaling to prove that weak solutions are locally continuous.

6.1 The Porous Medium Equation in its Own Geometry We assume the exponent γ satisfies the following assumptions:   (A1) γ ∈ L∞ 0, T ; W 1,p (Ω) , for some p > max{2, d}; (A2) For constants γ − , γ + > 0, 0 < γ − ≤ γ(x, t) ≤ γ + < ∞,

a.e. (x, t) ∈ ΩT .

88

6 Flows in Porous Media: The Variable Exponent Case

It is proved in [2], under less restrictive assumptions on γ, that there exists a unique solution to the initial boundary value problem associated with (6.1) and that the solution is bounded. It is also shown that the solution is nonnegative if the initial data is nonnegative and that is why it is reasonable to assume u(x, t) ∈ [0, 1] a.e. in ΩT . Here we consider local weak solutions. Definition 6.1. A measurable function u is a local weak solution of (6.1) if (i) u ∈ L∞ (0, T ; L∞ (Ω)) with u(x, t) ∈ [0, 1] a.e. in ΩT ;     γ(x,t) (ii) u ∈ C 0, T ; L2 (Ω) and u 2 ∇u ∈ L2 0, T ; L2 (Ω) ; (iii) for every compact K ⊂ Ω and for every subinterval [t1 , t2 ] ⊂ (0, T ], t2
t2

 −uφt + uγ(x,t) ∇u · ∇φ dx dt = 0, uφ dx + (6.2) K

t1

t1

K



   1 for all φ ∈ Hloc 0, T ; L2 (K) ∩ L2loc 0, T ; H01 (K) . As before we use the Steklov average to obtain the following formulation which is equivalent to (iii): (iii) for every compact K ⊂ Ω and for every 0 < t < T − h,
    (uh )t φ + uγ(x,·) ∇u · ∇φ dx = 0, h

K×{t}

(6.3)

for all φ ∈ H01 (K). We start by defining an intrinsic geometric configuration tailored for this specific PDE. Let (x0 , t0 ) be a point of the space-time domain ΩT that, by translation, we may assume to be (0, 0). Consider  small positive numbers > 0 and R > 0 such that the cylinder Q R2− , R ⊂ ΩT and define µ− :=

ess inf

Q(R2− ,R)

u;

µ+ := ess sup u ; Q(R2− ,R)

ω := ess osc u = µ+ − µ− . Q(R2− ,R)

Recalling that (6.1) is degenerate at the points where u = 0, the interesting case to investigate is when µ− = 0 and, consequently, µ+ = ω. From now on, we will assume this is in force. Construct the cylinder 2

Q(a0 R , R),

 γ + 4 a0 = , ω

and assume that 

ω ≥ 4R γ + .

(6.4)

6.2 Reducing the Oscillation





ess osc

u ≤ ω.

89

This implies that Q(a0 R2 , R) ⊂ Q R2− , R and then Q(a0 R2 ,R)

(6.5)

Remark 6.2. If (6.4) does not hold, then the oscillation ω goes to zero when the radius R goes to zero, in a way given by the reverse inequality, and there is nothing to prove. When γ ≡ 0, a0 = 1 and we recover the standard parabolic cylinder with the natural homogeneity of the space and time variables. Given ν0 ∈ (0, 1), to be determined in terms of the data and ω, either ω (6.6) (x, t) ∈ Q(a0 R2 , R) : u(x, t) < ≤ ν0 Q(a0 R2 , R) 2 or, noting that µ+ − ω2 = ω2 , ω (x, t) ∈ Q(a0 R2 , R) : u(x, t) > µ+ − < (1 − ν0 ) Q(a0 R2 , R) . 2

(6.7)

The analysis of this alternative leads to the following result. Proposition 6.3. There exist positive numbers ν0 , σ ∈ (0, 1), depending on the data and on ω, such that  ess osc2  ν R Q 20 a0 ( R 2 ) , 2

u ≤ σ ω.

(6.8)

An immediate consequence is the following. Theorem 6.4. Under assumptions (A1) − (A2) any locally bounded weak solution of (6.1) is locally continuous in ΩT . The proof follows from a slight modification of the arguments in Section 4.4. From (6.8) one defines recursively a sequence Qn of nested and shrinking cylinders and a sequence ωn converging to zero (see also the proof of Theorem 7.7 on Chapter 7), such that ess osc u ≤ ωn . Qn

This is enough to obtain the continuity of u but we are unable to derive a modulus since the constant σ appearing in Proposition 6.3 depends on the oscillation ω.

6.2 Reducing the Oscillation Assume that (6.6) is verified. In the following, we determine the number ν0 and guarantee that the solution u is above a smaller level within a smaller cylinder.

90

6 Flows in Porous Media: The Variable Exponent Case

Proposition 6.5. There exists ν0 ∈ (0, 1), depending only on the data and ω, such that if (6.6) holds then     2 R ω R , a.e. (x, t) ∈ Q a0 . (6.9) u(x, t) > , 4 2 2 Proof. This proof, as well as the subsequent ones, will be presented for the particular case p = ∞ in assumption (A1). The case of a general p > d is similar but technically more involved (see Remark 6.7). Define two decreasing sequences of positive numbers R R + n+1 , 2 2

Rn =

kn =

ω ω + n+2 , 4 2

n = 0, 1, . . .

and construct the family of nested and shrinking cylinders Qn = Q(a0 Rn2 , Rn ).  ω Introduce the function uω = max u, 4 . In the weak formulation (6.3) take φ = − ((uω )h − kn )− ξn2 , where 0 ≤ ξn ≤ 1 are smooth cutoff functions defined in Qn and satisfying ⎧ ξn ≡ 0 on the parabolic boundary of Qn ⎨ ξn ≡ 1 in Qn+1 , ⎩

|∇ξn | ≤

2n+2 R ,

|∆ξn | ≤

22(n+2) R2 ,

0 < (ξn )t ≤

22(n+2) a0 R2 ,

and integrate in time over (−a0 Rn2 , t), for t ∈ (−a0 Rn2 , 0). We obtain (omitting the dx and dt in all integrals from now on)
I1 + I2 :=




t

2 −a0 Rn
t

KRn

  (uh )t − ((uω )h − kn )− ξn2




+ 2 −a0 Rn

KRn

  (uγ ∇u)h · ∇ − ((uω )h − kn )− ξn2 = 0.

Concerning the first integral, we have

t   I1 = (uh )t − ((uω )h − kn )− ξn2 χ[(uω ) 2 −a0 Rn
t

KRn




+ =

1 2

2 −a0 Rn
t 2 −a0 Rn




KRn

KRn

h =uh

  (uh )t − ((uω )h − kn )− ξn2 χ[(uω ) 

2

((uω )h − kn )−

 t

ξn2

 
 ω 
t ω uh − + n+2 ξ2 . 2 4 − t n 2 −a0 Rn KRn

]

ω h= 4

]

6.2 Reducing the Oscillation

91

Next, we integrate by parts and let h → 0. Using Lemma 2.2, we get


t 1 2 2 (uω − kn )− ξn − (uω − kn )2− ξn (ξn )t 2 KRn ×{t} 2 −a0 Rn KRn +


   ω 
 ω 
t ω 2 ω u − u − ξ − 2 ξn (ξn )t 2n+2 KRn ×{t} 4 − n 2n+2 −a0 Rn2 KRn 4 −

 ω 2 22(n+2)
t 1 ≥ (uω − kn )2− ξn2 − 3 χ[uω ≤kn ] , 2 KRn ×{t} 4 a0 R2 −a0 Rn2 KRn

since the third term is nonnegative and, for 0 ≤ u ≤ ω 4 < u = uω ≤ kn , (uω − kn )− ≤ kn − uω = kn − u < kn −

ω 4,

uω =

ω 4

≤ kn and, for

ω ω ω = n+2 ≤ . 4 2 4

Concerning I2 , we first pass to the limit in h to get

t   I2 → uγ ∇u · ∇ −(uω − kn )− ξn2 2 −a0 Rn t




KRn




  uγ ∇u · ∇ −(uω − kn )− ξn2 χ[uω =u]

= 2 −a0 Rn
t

KRn




+


2 −a0 Rn

KRn




t

4

uγω |∇(uω − kn )− |2 ξn2

= 2 −a0 Rn
t

  uγ ∇u · ∇ −(uω − kn )− ξn2 χ[uω = ω ]

KRn




uγω ∇(uω − kn )− · ∇ξn ξn (uω − kn )−

+2 2 −a0 Rn

KRn 
t

 ω 2n+2

t




+2 ≥

1 2

−2

2 −a0 Rn
t 2 −a0 Rn

2 −a0 Rn

KRn

−uγ ∇u · ∇ξn ξn χ[u≤ ω ] 4

uγω |∇(uω − kn )− |2 ξn2 KRn




KRn

uγω |∇ξn |2 (uω − kn )2−


ω 
 ω 
t 4 +2 n+2 ∇ sγ ds · ∇ξn ξn χ[u≤ ω ] 4 2 2 −a0 Rn KRn u  

ω4  ω 
t (− ln s)sγ ds ∇γ · ∇ξn ξn χ[u≤ ω ] +2 n+2 4 2 2 −a0 Rn KRn u

92

6 Flows in Porous Media: The Variable Exponent Case

and this is bounded from below by

1 t  I2 := uγ |∇(uω − kn )− |2 ξn2 2 −a0 Rn2 KRn ω

t −2 uγω |∇ξn |2 (uω − kn )2− 2 −a0 Rn

 ω −2 n+2 2 − −

 ω 2n+2

KRn










t



ω 4

γ

s ds




2 −a0 Rn

t

2 −a0 Rn




KRn




KRn


 ω 
t 2n+2 −a0 Rn2 KRn

u

 (− ln s)sγ ds |∇γ|2 ξn χ[u≤ ω ] 4

u




4

u ω 4

  ξn |∆ξn | + |∇ξn |2 χ[u≤ ω ]

ω 4

 (− ln s)sγ ds |∇ξn |2 ξn χ[u≤ ω ] . 4

The first inequality is obtained by means of Cauchy’s inequality with = 14 , and the observation that, within the set [u ≤ ω4 ], 
ω  
ω  4

∇

sγ ds

4

= −uγ ∇u −

u

(− ln s)sγ ds ∇γ.

u

The second inequality is a consequence of integration by parts and again Cauchy’s inequality, this time with = 12 . Next, observe that for ω4 < u = uω ≤ kn , we get (uω − kn )− = kn − uω = kn − u < and For u ≤

 ω γ 1 ≤ < uγω ≤ 1 a0 4 ω 4,

we get

using

ω 2n+2

≤

ω 4

(A2).

ω ≤ kn ; 4
ω4  ω  ω γ  ω −u ≤ ; sγ ds ≤ 4 4 4 u ω ω
ω4

 ω γ 4 4 γ (− ln s)s ds ≤ (− ln s) ds ≤ (− ln s) ds 4 u u u    ω  4 ω ω ω  ω 4 + u ln u + − u ≤ −u ln +1 ≤ = 1. = − ln 4 4 4 4 ω 4 ω uω =

6.2 Reducing the Oscillation

93

Using these inequalities, recalling the conditions on ξn , and the fact that ω ≤ 1, I2 is bounded from below by
t
1 |∇(uω − kn )− |2 ξn2 2a0 −a0 Rn2 KRn


C(M )  ω 2 22(n+2) − ω 4 R2




t

2 −a0 Rn

KRn

χ[uω ≤kn ] ,

with M = γ L∞ (0,T ;W 1,∞ (Ω)) . Combining the above results, we obtain the energy estimates

0
1 2 2 (uω − kn )− ξn + |∇(uω − kn )− |2 ξn2 sup 2 µ+ − 2
− ν20 a0 R2 ω ≥ x ∈ KR : u(x, t) > µ+ − dt 2 −a R2  0    1 − ν0 ν0 a0 R2 ≥ |KR | 1 − ν0 1− 2 2   = (1 − ν0 ) Q a0 R2 , R  

contradicting (6.7).

Accordingly, at time level t0 , the portion of the cube KR where u(x) is close to its supremum is small. In what follows we will show that the same  happens for all time levels near the top of the cylinder Q a0 R2 , R . Lemma 6.9. There exists s1 ∈ N, depending on the data and ω, such that, for all t ∈ (t0 , 0),   ν 2  ω 0 |KR | . (6.14) x ∈ KR : u(x, t) > µ+ − s1 < 1 − 2 2 Proof. Consider the cylinder Q (t0 , R), the level k = µ+ − c=

ω 2n∗ +1

ω 2

and put

(n∗ to be chosen).

Define u − k ≤ Hk+ := ess sup (u − k)+ ≤ Q(t0 ,R)

ω . 2

If Hk+ ≤ ω4 , the result is trivial for the choice s1 = 2. Assuming Hk+ > logarithmic function ψ + given by ⎧   Hk+ ⎪ if u > k + c ⎨ ln H + −u+k+c k ψ+ = ⎪ ⎩ 0 if u ≤ k + c

ω 4,

the

is well defined in Q (t0 , R) and satisfies the inequalities ψ + ≤ ln(2n∗ ) = n∗ ln 2,

since

Hk+ Hk+ ≤ ≤ c Hk+ − u + k + c

ω 2 ω 2n∗ +1

= 2n∗

6.3 Analysis of the Alternative

97

and, for u = k + c,   0 ≤ ψ+ ≤ and



2n∗ +1 1 1 ≤ = c ω −u+k+c

Hk+

ψ+



=



ψ+

  2

≥ 0.

In the weak formulation (6.2) [to be rigorous we should consider, as before, formulation (6.3), integrate and take the limit as h → 0; to simplify we proceed formally at this stage] consider the integration over KR × (t0 , t), with t ∈  (t0 , 0), and take φ = 2ψ + (ψ + ) ξ 2 , where x → ξ(x) is a smooth cutoff function defined in KR and verifying ⎧ ⎪ ⎪ 0 ≤ ξ ≤ 1 in KR ; ⎪ ⎪ ⎨ ξ ≡ 1 in K(1−σ)R , for some σ ∈ (0, 1); ⎪ ⎪ ⎪ ⎪ ⎩ C . |∇ξ| ≤ σR Then, for all t ∈ (t0 , 0),
t


  ut 2ψ + ψ + ξ 2 +

0 = t0


t
t0

KR

    uγ(x,t) ∇u · ∇ 2ψ + ψ + ξ 2

KR

=: J1 + J2 . The two integrals can be estimated as follows:

 + 2 2 ψ ξ − J1 = KR ×{t}


≥



ψ+

2

KR ×{t0 }



ψ

 + 2

KR ×{t}



ξ − 2

n2∗

2

ln 2

1 − ν0 1 − ν20

ξ2  |KR |,

using the estimate for ψ + and Lemma 6.8;
t
   +   2 2  J2 = ψ uγ(x,t) |∇u|2 2 1 + ψ + ξ t0

KR


t


  uγ(x,t) ∇u · ∇ξ 2ψ + ψ + ξ

+2 t0


t


KR

uγ(x,t) |∇ξ|2 2ψ + ,

≥− t0

KR

using Cauchy’s inequality. Putting these estimates together, using the bounds for |∇ξ| and ψ + , and recalling the values of t0 and a0 , we arrive at

98

6 Flows in Porous Media: The Variable Exponent Case






ψ

 + 2

KR ×{t}

   1 − ν0 2 2 + 2n∗ ln 2 ξ ≤ n∗ ln 2 1 − ν20    1 − ν0 + 2n∗ ln 2 ≤ n2∗ ln2 2 1 − ν20    1 − ν0 ≤ n2∗ ln2 2 + 2n∗ ln 2 1 − ν20 2

 C (−t0 ) |KR | σ 2 R2  C a0 |KR | σ2  C(γ + ) |KR | , σ2 ωγ+

valid for all t ∈ (t0 , 0). The left hand side is estimated from below integrating over the smaller set  ω  S = x ∈ K(1−σ)R : u(x, t) > µ+ − n∗ +1 . 2 On S, ξ ≡ 1 and ψ + ≥ (n∗ − 1) ln 2, because Hk+ Hk+ ≥ Hk+ − u + k + c Hk+ − ω2 +

ω 2n ∗

=

Hk+ − ω2 + ω2 ≥ 2n∗ −1 , Hk+ − ω2 + 2ωn∗

since one has Hk+ − ω2 ≤ 0 and ω2 > 2ωn∗ , ∀n∗ > 1. Therefore, for all t ∈ (t0 , 0),   2   1 − ν0 C n∗ |S| ≤ + 2 |KR | . n∗ − 1 1 − ν20 σ n∗ ω γ + Consequently, for all t ∈ (t0 , 0), x ∈ KR : u(x, t) > µ+ −

≤ |S| + dσ |KR | 2n∗ +1   2   1 − ν0 C n∗ + 2 + dσ |KR | . ≤ n∗ − 1 1 − ν20 σ n∗ ω γ + ω

The proof is complete once we choose σ so small that dσ ≤ 38 ν02 , then n∗ so large that 3 2 C ν + ≤ 2 γ 8 0 n∗ σ ω

 and

n∗ n∗ − 1

2

 ν0  (1 + ν0 ) =: β > 1, ≤ 1− 2  

and finally take s1 = n∗ + 1. Remark 6.10. Note that, from the choice of σ, we get σ < two conditions on n∗ we obtain

 4 −6 −γ + n∗ ≥ max Cν0 ω ; 2 +2 . ν0

3 2 8d ν0

Clearly the number s1 depends on the data as well as on ω.

and from the

6.3 Analysis of the Alternative

99

  Recalling that t0 ∈ −a0 R2 , − ν20 a0 R2 , the next result follows from the previous lemma. Corollary 6.11.  s1 ∈ N, depending on the data and ω, such  There exists that, for all t ∈ − ν20 a0 R2 , 0 ,   ν 2  ω 0 + |KR |. (6.15) x ∈ KR : u(x, t) > µ − s1 < 1 − 2 2 The conclusion 6.11 will be employed to deduce that, within   of Corollary the cylinder Q ν20 a0 R2 , R , the set where u is close to its supremum is arbitrarily small. Lemma 6.12. For all ν ∈ (0, 1) there exists s1 < s2 ∈ N, depending on the data and ω, such that ν   ν ω 0 0 a0 R2 , R : u(x, t) > µ+ − s2 ≤ ν Q a0 R2 , R . (x, t) ∈ Q 2 2 2   Proof. Consider the cylinder Q ν0 a0 R2 , 2R and the levels k = µ+ − 2ωs , for s ≥ s1 . In order to obtain energy estimates for the functions (u − k)+ over 2 this cylinder, we take φ = (u  − k)+2ξ in (6.2), where 0 ≤ ξ ≤ 1 is a smooth cutoff function defined in Q ν0 a0 R , 2R and satisfying   ⎧ ξ ≡ 0 on ∂p Q ν0 a0 R2 , 2R ; ⎪ ⎪ ⎪ ⎪ ⎨   ξ ≡ 1 in Q ν20 a0 R2 , R ; ⎪ ⎪ ⎪ ⎪ ⎩ |∇ξ| ≤ 1 ; 0 ≤ ξt ≤ ν0 a1 R2 . R 2

0





Then, for t ∈ −ν0 a0 R2 , 0 , we obtain (again formally)





t

−ν0 a0 R2


ut (u − k)+ ξ 2 +

K2R

Now, since (u − k)+ ≤


t

−ν0 a0 R2

−ν0 a0 R2

  uγ(x,t) ∇u · ∇ (u − k)+ ξ 2 = 0.

K2R

ω 2s ,


ut (u − k)+ ξ 2 = K2R




t

1 2 −

≥−


(u − k)2+ ξ 2

K2R ×{t}

t −ν0 a0 R2

 ω 2 2s

(u − k)2+ ξ ξt K2R

1 ν0 2 2 a0 R




t

−ν0 a0 R2


χ[u>k] K2R

100

6 Flows in Porous Media: The Variable Exponent Case

and

≥




1 2a0







t −ν0 a0 R2

K2R




t

−ν0 a0 R2

  uγ(x,t) ∇u · ∇ (u − k)+ ξ 2

|∇(u − k)+ |2 ξ 2 − 2 K2R


 ω 2 1
t χ[u>k] , 2s R2 −ν0 a0 R2 K2R

using Cauchy’s inequality with = 14 and the fact that when (u − k)+ is not zero, then ω ω ω ω u > k = µ+ − s > µ+ − = > , 2 2 2 4 since s ≥ s1 > 1, and therefore 1 ≥ uγ(x,t) ≥ We then have 1 2a0

 ω γ + 4

=

1 . a0



ν0 2

|∇(u − k)+ |2

Q( a0 ) 

 ω 2 1  1 ≤ s +2 χ[u>k] 2 R2 ν20 a0 Q(ν0 a0 R2 ,2R)  ν  ω 2 1  1  0 d+1 2 a ≤ s + 2 2 R , R Q 0 2 R2 ν20 a0 2 R2 ,R

and consequently, multiplying by 2a0 ,  ν  ω 2 1  4  0 d+1 2 a 2 + 4a Q R , R |∇(u − k)+ | ≤ 0 0 s 2 ν0 2 R ν0 2 Q( 2 a0 R2 ,R) ν  2d+3  ω 2 1 0 2 a (1 + a ν ) Q R , R = 0 0 0 ν0 2s R2 2  C(M, d, γ + )  ω 2 1  ν0 a0 R2 , R , ≤ Q (d+2) s 2 + (1+γ ) 2 R 2 2 ω





2

recalling the definition of ν0 given by (6.10). Now we consider the levels ω ω > k1 = µ+ − s , 2s+1 2   and define, for t ∈ − ν20 a0 R2 , 0 , k2 = µ+ −

s = s1 , . . . , s2 − 1,

 ω As (t) := x ∈ KR : u(x, t) > µ+ − s 2

6.3 Analysis of the Alternative

and


As :=

101

0

−

ν0 2

|As (t)| dt. a0 R2

  Using Lemma 2.7, applied to the function u(·, t) for all times t ∈ − ν20 a0 R2 , 0 , we get
 ω  Rd+1 |As+1 (t)| ≤ C(d) |∇u|. 2s+1 |KR \ As (t)| [k1 kn ] , I1 ≥ ν0 2 ν 2 KRn ×{t} 2 2 − 20 a0 Rn KRn 2 a0 R

6.3 Analysis of the Alternative

and 1 I2 ≥ 2




−2




t

−




103

ν0 2

uγ(x,t) |∇(uω − kn )+ |2 ξn2 ω

2 a0 Rn

KRn




t ν − 20

2 a0 Rn

uγ(x,t) (uω − kn )2+ |∇ξn |2 ω KRn





 ω 
t −2 s2 +1 ν 2 2 − 20 a0 Rn KRn
 ω 
t − s2 +1 ν 2 2 − 20 a0 Rn KRn



u γ(x,t)

s µ+ −

ds

ω 2s2 +1

  |∆ξn | + |∇ξn |2 χ u≥µ+ − ω  2s2 +1 
 u

sγ(x,t) (− ln s) ds 

µ+ −

ω 2s2 +1

 |∇γ|2 + |∇ξn |2 χ u≥µ+ −

ω 2s2 +1

,

and, ultimately, the estimate


1 2 (uω − kn )+ ξn2 + |∇ (uω − kn )+ |2 ξn2 sup ν0 a 2 0 Qn − a0 Rn 0 and let γ(s) = s + λH(s), where λ is a positive constant (physically, the latent heat of the phase transition):

7.1 Regularization of the Maximal Monotone Graph γ

109

6

λ

-

Let 0 <  1 and consider the function γ (s) = s + λH (s), where H is a C ∞ approximation of the Heaviside function, such that H (s) = 0

if

s≤0;

H (s) = 1 if

s ≥ ,

H (s) ≥ 0, s ∈ R and H −→ H uniformly in compact subsets of R \ {0} as → 0. The function γ is Lipschitz continuous, together with its inverse, and satisfies (7.4) 1 ≤ γ (s) ≤ 1 + λL , s ∈ R, where L ≡ O( 1 ) is the Lipschitz constant of H . Its inverse β = γ−1 satisfies 0
max AR 2−p , BR 2(2−p) . 1

Then Q(a0 Rp , c0 R) ⊂ Q(R, R 2 ) and ess osc

Q(a0 Rp ,c0 R)

θ ≤ ω.

(7.15)

We now consider the cube Kc0 R partitioned in disjoint subcubes, each congruent to Kd∗ R , with d∗ = ( 2nω∗ +1 )p−2 , for n∗ to be determined: [¯ x + Kd∗ R ] ,

x ¯ ∈ KR(ω) , 

L1 =

B 2n∗ +1

R(ω) := c0 R − d∗ R = L1 d∗ R

2−p − 1,

B > 2n∗ +1 .

Since we may arrange L1 to be an integer, we can look at Kc0 R as the disjoint union, up to a set of measure zero, of Ld1 cubes of the above type. Then we

114

7 Phase Transitions: The Doubly Singular Stefan Problem

may regard the cylinder Q(a0 Rp , c0 R) as the disjoint union, up to a set of x, 0) + Q(a0 Rp , d∗ R)]. measure zero, of Ld1 subcylinders of the type [(¯ We next consider subcylinders of [(¯ x, 0) + Q(a0 Rp , d∗ R)] of the form [(¯ x, t¯) + Q(d∗ Rp , d∗ R)],

d∗ =

 ω (1−p)(2−p) 2

,

which are contained in [(¯ x, 0) + Q(a0 Rp , d∗ R)] assuming that A > 2 and t¯ ∈ I(ω) := (−a0 Rp + d∗ Rp , 0) .

(7.16)

The proof of the equicontinuity relies on the study of two complementary cases and the achievement of the same conclusion for both, namely the reduction of the oscillation. For a given constant ν0 ∈ (0, 1), which will be determined later only in terms of the data and ω, we assume that either The First Alternative there exists t¯ ∈ I(ω) such that, for all x ¯ ∈ KR(ω) , (x, t) ∈ [(¯ x, t¯) + Q(d∗ Rp , d∗ R)] : θ(x, t) < µ− + ω2 ≤ ν0 |Q(d∗ Rp , d∗ R)|

(7.17)

or The Second Alternative for every t¯ ∈ I(ω), there exists x ¯ ∈ KR(ω) such that (x, t) ∈ [(¯ x, t¯) + Q(d∗ Rp , d∗ R)] : θ(x, t) > µ+ − ω2 < 1 − ν0 . |Q(d∗ Rp , d∗ R)|

(7.18)

In the first part of the alternative, we deal with the singularity in time (degeneracy in space) so what dominates the geometry is the scaling in time; the type of partition of the cylinders that is considered is a reflection of this fact. In the second part of the alternative, everything takes place above the singularity in time, the singular character of the principal part thus being the dominant factor; it comes with no surprise that the type of partition considered there is a partition in space. As a consequence of the alternative, we obtain a constant σ = σ(ω) ∈ (0, 1), depending only on the data and ω, such that ess osc

p R Q(d∗ ( R 8 ) ,c0 8 )

θ ≤ σ(ω)ω.

7.3 The Intrinsic Geometry

115

As usually, we next define recursively two sequences of real positive numbers. Let R , and R1 = ω1 = σ(ω)ω C(ω1 ) where   (p−1)(2−p) p (1−p)(2−p) p−2 A p 8σ(ω) B(ω1 )2−p σ(ω) p > 8. C(ω1 ) = 2 Defining Q1 =

(a1 R1p , c1 R1 )

;

with a1 =

 ω (1−p)(2−p) 1

A

 ,

c1 =

ω1 B(ω1 )

p−2

and noting that  ω (1−p)(2−p)

Rp A C(ω1 )p  ω (1−p)(2−p)  R p σ(ω)2−p = 2 8 B(ω1 )p(2−p)  p R ≤ d∗ 8

a1 R1p =

1

and p−2 ω1 R B(ω1 ) C(ω1 )  ω p−2  R  1 = (p−1)(2−p)   B 8 p B 2−p A 2 R ≤ c0 8 

c1 R1 =

we get

 Q1 ⊂ Q d

∗



R 8

p

R , c0 8



and, consequently, ess osc θ ≤ Q1

ess osc p

R Q(d∗ ( R 8 ) ,c0 8 )

θ ≤ σ(ω)ω = ω1 .

The process can now be repeated starting from Q1 since (7.15) holds in this cylinder. We then define the following recursive sequences of real positive numbers ⎧ ⎧ ⎨ R0 = R ⎨ ω0 = ω and ⎩R ⎩ Rn ωn+1 = σ(ωn )ωn n+1 = C(ωn+1 )

116

7 Phase Transitions: The Doubly Singular Stefan Problem

and construct the family of nested and shrinking cylinders Qn = (an Rnp , cn Rn ),

n = 0, 1, . . .

with an =

 ω (1−p)(2−p) n

A

 ,

cn =

ωn B(ωn )

p−2 .

Theorem 7.7. The sequences (ωn )n and (Rn )n are decreasing sequences converging to zero. Moreover, for every n = 0, 1, . . . Qn+1 ⊂ Qn

and

ess osc θ ≤ ωn . Qn

(7.19)

Proof. The sequences are obviously decreasing and bounded below by zero, so to show that they converge to zero we just need to show that they cannot converge to a positive number. As far as (Rn )n is concerned this conclusion follows immediately from 1 1 Rn+1 < . = Rn C(ωn+1 ) 8 With (ωn )n the situation is more delicate since ωn+1 = σ(ωn )  1, ωn due to the special form of σ (see Corollaries 7.13 and 7.23). So we suppose that ωn  α > 0 and observe that, in that case, σ(ωn )  σ(α) < 1. Consequently, ωn+1 = σ(ωn ) ωn ≤ σ(α) ωn and ωn ≤ [σ(α)]

n

ω,

n = 0, 1, . . . ,

which implies that ωn → 0, a contradiction. Relations (7.19) follow at once from the recursive process used to define the sequences.   The next result is now standard (see Section 4.4). Theorem 7.8. The sequence (θ ) is locally equicontinuous, i.e., for each θ , there exists an interior modulus of continuity that is independent of . Therefore equation (7.1) has, at least, one locally continuous solution.

7.4 Analyzing the Singularity in Time

117

7.4 Analyzing the Singularity in Time Assume that (7.17) holds in [(¯ x, t¯) + Q(d∗ Rp , d∗ R)], where [¯ x + Kd∗ R ] is any subcube of the partition of Kc0 R and t¯ is the same in all these cylinders. Since Kc0 R is the disjoint union, up to a set of measure zero, of subcubes of the form [¯ x + Kd∗ R ], the first alternative implies that there exists a cylinder of the type [(0, t¯) + Q(d∗ Rp , c0 R)] in which (x, t) ∈ [(0, t¯) + Q(d∗ Rp , c0 R)] : θ(x, t) < µ− + ω 2 ≤ ν0 . (7.20) |Q(d∗ Rp , c0 R)| Lemma 7.9. There exists a constant ν0 ∈ (0, 1), depending only on the data and ω, such that if (7.17) holds for some t¯ ∈ I(ω) and all x ¯ ∈ KR(ω) , θ(x, t) > µ− +

ω , 4

   p  R R a.e. (x, t) ∈ (0, t¯) + Q d∗ . , c0 2 2

(7.21)

Proof. According to the remark preceding the statement of the lemma, (7.20) is in force. After translation, we may assume that t¯ = 0. Define two decreasing sequences of numbers Rn =

R R + n+1 , 2 2

kn = µ− +

ω ω + n+2 ; 4 2

n = 0, 1, 2, . . .

and construct the family of nested and shrinking cylinders Qn = Q(d∗ Rnp , c0 Rn ). Write the energy estimate (7.9) for the functions (θ − kn )− , over Qn , and for smooth cutoff functions 0 ≤ ξn ≤ 1, defined in Qn , and such that ⎧ ⎨ ξn ≡ 1 in Qn+1 ; ξn ≡ 0 on ∂p Qn ⎩ Since (θ − kn )− ≤

|∇ξn | ≤

ω 2,

2n+2 c0 R

;

0 < (ξn )t ≤








Kc0 Rn ×{t}

 ω p 2(n+2)p ≤C 2 cp0 Rp ≤C

.

we get

sup p −d∗ Rn ν˜0 [ψk (θ) 0 using (7.28) for k = − 21 and ν˜0 = Theorem 2.12 to conclude that

¯ p ξ p ≥ C ∇ψk (θ) K4

 ν 2  2s2 d(2−p) 0

2

K4

B

¯ p, ψkp (θ)ξ

> 0. We can then apply

C = C(d, p, ν˜0 )

and consequently the following result. Lemma 7.18. There exist constants C0 and C1 , that can be determined a priori only in terms of, respectively, p, d, ω, n∗ , B and the data, and p, d, n∗ , B, such that

d ¯ p + C0 ¯ p ≤ C1 . φk (θ)ξ ψkp (θ)ξ (7.35) dt K4 K4 This integral inequality will be used to prove an auxiliary proposition which is an important tool for what follows. Introduce the quantities
Yn := sup ξp, n = 0, 1, . . . (7.36) −4p ≤t≤0

n] ¯ K4 ∩[θ>−|δ|

Proposition 7.19. The number ν being fixed, we can find numbers δ, σ, depending only on the data, ω and ν, and independent of , such that, for n = 0, 1, 2, . . ., either Yn ≤ ν or Yn+1 ≤ max{ν, σYn }.

134

7 Phase Transitions: The Doubly Singular Stefan Problem

Proof. Take k = −|δ|n in (7.35), where δ ∈ (−1, 0) is to be chosen. From (7.36), it follows that for every ρ ∈ (0, 1) there exists t0 ∈ (−4p , 0) such that
Yn+1 − ρ ≤ ξ p (·, t0 ), n = 0, 1, 2, . . . (7.37) n+1 ] ¯ K4 ∩[θ>−|δ|

After fixing n ∈ N and t0 ∈ (−4p , 0), one of the following two situations holds: either 
 d ¯ p (t0 ) ≥ 0 φ−|δ|n (θ)ξ (7.38) dt K4 or 
 d p ¯ φ−|δ|n (θ)ξ (t0 ) < 0. (7.39) dt K4 In either case we may assume that Yn > ν (otherwise the result is trivial). Assume that (7.38) holds. Then
p ¯ p ψ−|δ| n (θ)ξ (·, t0 ) ≤ C, K4

for a positive constant C, independent of , where     |δ|n (1 − δ) 1−δ p p p ¯ . ≥ ln ( θ(x, t )) = ln ψ−|δ| n 0 ¯ t0 ) −2δ −|δ|n δ − θ(x, Perform an integration over the smaller set

 ¯ t0 ) > −|δ|n+1 x ∈ K4 : θ(x, to get


¯ 0 )>−|δ|n+1 ] K4 ∩[θ(.,t

ξ p (·, t0 ) ≤ C ln−p



1−δ −2δ

 .

Using (7.37) and taking, without loss of generality, ρ ∈ (0, ν2 ), we obtain   1−δ ν −p Yn+1 ≤ + C ln . 2 −2δ We take |δ| so small that −p

C ln that is |δ| = −δ ≤



1−δ −2δ

 ≤

ν , 2

1 1 p 2 exp ( 2C ν )

−1

∈ (0, 1)

and the proposition is proved assuming that (7.38) holds.

7.5 The Effect of the Singularity in the Principal Part

135

Now assume that (7.39) holds and define 
 

d ¯ p ≥0 . φ−|δ|n (θ)ξ t∗ := sup t ∈ (−4p , t0 ) : dt K4


Then

K4

K4

¯ p (·, t∗ ) φ−|δ|n (θ)ξ

 ds ξ p (·, t∗ ) χ[(θ+|δ| = n ) >s] ¯ + (|δ|n (1 − δ) − s)p−1 K4 0 

1 |δ|n(2−p) = χ[(θ+|δ| ds ξ p (·, t∗ ) n ) >s|δ|n ] ¯ + p−1 (1 − δ − s) 0 K4 

1 |δ|n(2−p) p ξ (·, t∗ ) ds. = p−1 n ) >s|δ|n ] ¯ 0 (1 − δ − s) K4 ∩[(θ+|δ| +






¯ p (·, t0 ) ≤ φ−|δ|n (θ)ξ |δ|n

We estimate from above the integral in brackets, for s ∈ [0, 1]. On the one hand we have, using the definition of Yn ,

ξ p (·, t∗ ) ≤ ξ p (·, t∗ ) ≤ Yn . n ) >s|δ|n ] ¯ K4 ∩[(θ+|δ| +

n] ¯ K4 ∩[θ>−|δ|

On the other hand, from the definition of t∗ , we first get
¯ p (·, t∗ ) ≤ C ψ p n (θ)ξ K4

−|δ|

and then, integrating over the smaller set K4 ∩ [(θ¯ + |δ|n )+ > s|δ|n ], we obtain
ξ (·, t∗ ) ≤ C ln p

n ) >s|δ|n ] ¯ K4 ∩[(θ+|δ| +

since, in this set,

 p p ¯ ψ−|δ| n (θ) ≥ ln

Then, for all s ∈ [0, 1],


1−δ 1−δ−s



1−δ 1−δ−s



 .

−p

ξ (·, t∗ ) ≤ min Yn , C ln p

n ) >s|δ|n ] ¯ K4 ∩[(θ+|δ| +

−p



1−δ 1−δ−s

 .

136

7 Phase Transitions: The Doubly Singular Stefan Problem

Let s∗ be such that Yn = C ln−p



1−δ 1−δ−s∗

 , i.e.,

1

s∗ = For 0 ≤ s < s∗ −p



C ln

and for s∗ ≤ s ≤ 1 C ln Then




1

0


≤

s∗

0

−p



exp ( YCn ) p − 1 1

exp ( YCn ) p

1−δ 1−δ−s

1−δ 1−δ−s

|δ|n(2−p) (1 − δ − s)p−1



−p

(1 − δ).



> C ln

 ≤ C ln

−p



1−δ 1 − δ − s∗

1−δ 1 − δ − s∗

 = Yn  = Yn .




 p

n ) >s|δ|n ] ¯ K4 ∩[(θ+|δ| +

|δ|n(2−p) Yn ds + (1 − δ − s)p−1




1 s∗




ξ (·, t∗ )

|δ|n(2−p) C ln−p (1 − δ − s)p−1



ds 1−δ 1−δ−s

 ds

1

1 ds (1 − δ − s)p−1 0  

 C −p 1−δ ds . 1− ln Yn 1−δ−s

= |δ|n(2−p) Yn


1

− s∗

1 (1 − δ − s)p−1

Our next goal is to obtain an estimate from below, independent of Yn , for the second integral on the right hand side of this inequality. We start by noting that 1 exp ( Cν ) p − 1 σ0 := s∗ < σ0 (1 − δ), 1 exp ( Cν ) p since we are assuming that Yn > ν, and for s∗ ≤ s ≤ 1   1−δ C −p . ln 0≤1− Yn 1−δ−s Therefore




1

s∗




1

≥ σ0 (1−δ)






 C −p 1−δ ds 1− ln Yn 1−δ−s 

 C −p 1−δ 1 ds 1− ln (1 − δ − s)p−1 Yn 1−δ−s 

 C −p 1−δ 1 ds. 1 − ln (1 − δ − s)p−1 ν 1−δ−s

1 (1 − δ − s)p−1

1

≥ σ0 (1−δ)

7.5 The Effect of the Singularity in the Principal Part

We obtain





1

1 ds p−1 0 (1 − δ − s) K4  !


1 C −p 1−δ 1 ds 1 − ln − p−1 ν 1−δ−s σ0 (1−δ) (1 − δ − s) 

1−|δ| 1 1 1 n(2−p) = |δ| Yn ds − − ds p−1 (1 − δ − s) (1 − δ − s)p−1 0 1−|δ|  !


1 C −p 1−δ 1 ds 1 − ln + p−1 ν 1−δ−s σ0 (1−δ) (1 − δ − s) ¯ p (·, t0 ) ≤ |δ|n(2−p) Yn φ−|δ|n (θ)ξ


= |δ|

n(2−p)

1−|δ|

Yn [1 − f (δ)] 0

1 ds (1 − δ − s)p−1

where


1−|δ|

f (δ) 0




1

+ σ0 (1−δ)

1 ds = − (1 − δ − s)p−1

1 (1 − δ − s)p−1




1 1−|δ|

1 ds (1 − δ − s)p−1



 C −p 1−δ ds. 1 − ln ν 1−δ−s

To get a lower bound on f (δ) note that   1−δ C −p ≥ 0; (i) for σ0 (1 − δ) ≤ s ≤ 1, 1 − ln ν 1−δ−s 1 p exp ( 2C ν ) −1 (ii) σ0 ≤ σ1 := ; 1 p exp ( 2C ν )   1 1−δ C −p ≥ . (iii) for σ1 (1 − δ) ≤ s ≤ 1, 1 − ln ν 1−δ−s 2 Then


1−|δ|

1 ds (1 − δ − s)p−1 0

1 1 1 1 1 ds + ds ≥− p−1 2 σ1 (1−δ) (1 − δ − s)p−1 1−|δ| (1 − δ − s)   1 1 1 (−δ)2−p + (1 − δ)2−p (1 − σ1 )2−p − (−2δ)2−p = 2−p 2 2   1 1 (1 − δ)2−p (1 − σ1 )2−p − (−2δ)2−p ≥ 2−p 2 f (δ)

137

138

7 Phase Transitions: The Doubly Singular Stefan Problem

and consequently 1 f (δ) ≥ (1 − σ1 )2−p − 2



−2δ 1−δ

2−p .

Choosing δ ∈ (−1, 0) such that 2−p  1 −2δ = (1 − σ1 )2−p 1−δ 4 we get 1 (1 − σ1 )2−p 4 and then, for σ := 1 − 14 (1 − σ1 )2−p ∈ (0, 1), f (δ) ≥





K4

¯ p (·, t0 ) ≤ σYn φ−|δ|n (θ)ξ

1−|δ|

0

|δ|n(2−p) ds. (1 − δ − s)p−1

To estimate from below the integral over K4 , we integrate over the smaller ¯ t0 ) > −|δ|n+1 ] to get, using (7.37), set K4 ∩ [θ(·,

p ¯ ¯ p (·, t0 ) φ−|δ|n (θ)ξ (·, t0 ) ≥ φ−|δ|n (θ)ξ K4




¯ 0 )>−|δ|n+1 ] K4 ∩[θ(.,t

 1 = ds ξ p (·, t0 ) (|δ|n (1 − δ) − s)p−1 ¯ 0 )>−|δ|n+1 ] 0 K4 ∩[θ(.,t 
 
 1−|δ| |δ|n(2−p) p ≥ ξ (·, t0 ) ds (1 − δ − s)p−1 ¯ 0 )>−|δ|n+1 ] 0 K4 ∩[θ(·,t
1−|δ| |δ|n(2−p) ≥ (Yn+1 − ρ) ds. (1 − δ − s)p−1 0


|δ|n +θ¯

This and the previous estimate yield, since ρ is arbitrary in (0, ν2 ), Yn+1 ≤ σYn , which completes the proof also for the case that (7.39) holds. We now remove assumption (7.30). If (7.30) does not hold we have to make use of the discrete time derivative in order to obtain the weak formulation of (7.29). This means that, for all t ∈ [−4p + h, 0], and h > 0 we have
t

t
¯ ¯ p−2 ∇θ¯ · ∇ϕ = 0, ∂t (θ)ϕ + C |∇θ| t−h

K4

t−h

K4

for all admissible testing functions ϕ, where C is a positive constant independent of . Being (θ¯ − k)+ , with k ∈ (−1, 0), a sub-solution of (7.29) and taking ϕ as before, we obtain, for all t ∈ [−4p + h, 0] and h > 0,

7.5 The Effect of the Singularity in the Principal Part


C2 h ≥







K4 ×{t}

¯ p− φk (θ)ξ

¯ p + C1 φk (θ)ξ

K4 ×{t−h}

t

139




t−h

K4

¯ p. ψkp (θ)ξ

Dividing by h and letting h → 0 we get an integral inequality similar to (7.35): 

d dτ

−



¯ p + C1 φk (θ)ξ

K4

K4

¯ p ≤ C2 ψkp (θ)ξ

where !  −


d 1 ¯ p := lim sup ¯ p− ¯ p . φk (θ)ξ φk (θ)ξ φk (θ)ξ dτ h→0 h K4 K4 ×{t} K4 ×{t−h} Define the set



S :=

 t ∈ (−4 , 0) : p

d dτ

−


 ¯ φk (θ)ξ ≥0 p

K4

/ S, and let t0 be given as in (7.37). If t0 ∈ S, we proceed as in (7.38). If t0 ∈ take t¯ = sup{t ∈ (−4p , t0 ) : t ∈ S} ≤ t0 . If t¯ = t0 , consider a sequence (tn )n , tn ∈ S, such that tn → t0 . Since tn ∈ S we get
¯ p ≤ C. ψkp (θ)ξ K4 ×{tn }




Then

K4 ×{t0 }

¯ p≤C ψkp (θ)ξ

and we proceed as in (7.38). If t¯ < t0 we have ⎧
p ¯ p ⎪ ⎪ ⎪ ⎪ K ×{t¯} ψk (θ)ξ ≤ C ⎨ 4
⎪ ⎪ ⎪ ⎪ ⎩


¯ p (x, t0 ) ≤ φk (θ)ξ

K4

¯ p (x, t¯) φk (θ)ξ K4

 

and we reason as in (7.39). 7.5.2 Expansion in Space From Proposition 7.19, we get by iteration Yn ≤ max{ν, σ n Y0 },

n = 1, 2, . . .

140

7 Phase Transitions: The Doubly Singular Stefan Problem




and since Y0 =

sup

−4p ≤t≤0

¯ K4 ∩[θ(.,t)>−1]

ξ p (·, t) ≤ |K4 |

we obtain Yn ≤ max{ν, σ n |K4 |} = max{ν, σ n 2d |K2 |},

n = 1, 2, . . .

Take n = n0 ∈ N so large that σ n0 2d ≤ ν. Then Yn0 ≤ max{ν, σ n0 2d |K2 |} ≤ max{ν, ν|K2 |} = ν|K2 |. Recalling the definition of Yn , as well as the choice of ξ, we obtain
¯ t) > −|δ|n0 , Yn0 ≥ sup ξ p (·, t) ≥ x ∈ K2 : θ(x, −4p ≤t≤0

n0 ] ¯ K2 ∩[θ(.,t)>−|δ|

for all t ∈ [−2p , 0], and therefore ¯ t) > −|δ|n0 ≤ ν|K2 |, x ∈ K2 : θ(x,

∀t ∈ [−2p , 0].

We have just proven the crucial result towards the expansion to a full cylinder in space. Lemma 7.20. Given ν ∈ (0, 1), there exists δ ∗ ∈ (0, 1), depending only on the data, ν and ω, such that ¯ t) > −δ ∗ ≤ ν|K2 |, x ∈ K2 : θ(x,

∀t ∈ [−2p , 0].

(7.40)

To prove the main result of this section, we need to make use of another auxiliary result, which proof is a trivial modification to sub-solutions of the proof of Lemma 4.1 of [14, Ch. 4]. Indeed, being above the singularity in time, we are dealing only with powers p and 2 in the energy estimates. Lemma 7.21. There exists ν˜, depending on the data, d and p, and independent of ω and , such that if θ is a sub-solution of (7.6) in [(¯ x, t¯) + QR (m1 , m2 )] satisfying ess osc

[(¯ x,t¯)+QR (m1 ,m2 )]

and

θ≤ω

ω x, t¯) + QR (m1 , m2 )] : θ(x, t) > µ+ − m ≤ ν˜ |QR (m1 , m2 )| (x, t) ∈ [(¯ 2

then θ(x, t) ≤ µ+ −

ω

, 2m+1

  ∀(x, y) ∈ (¯ x, t¯) + Q R (m1 , m2 ) , 2

7.5 The Effect of the Singularity in the Principal Part

141

where m = m1 + m2 , m1 , m2 ≥ 0 and  ω  p−2 p . 2m1 We are now in a position to prove the main result of this section.   QR (m1 , m2 ) = Kd1 R × −2m2 (p−2) Rp , 0 ,

d1 =

Proposition 7.22. Assume the second alternative holds. There exists s3 > s2 such that   p  ω a0 R + θ(x, t) ≤ µ − s3 , a.e. (x, t) ∈ Q , c0 R . (7.41) 2 2 4 Proof. Note that we are done if we prove that (7.41) holds in the cylinder    p    p  a0 R a0 R (¯ x, 0) + Q , 2c0 R ⊇Q , c0 R , 2 4 2 4 independently of the location of x ¯ in KR(ω) . We will prove that there exists s4 > 1 such that ¯ t) ≤ − 1 , θ(x, 2s4

a.e. (x, t) ∈ Q1 = K1 × (−1, 0),

and then use the fact that t¯ is an arbitrary element of I(ω) to get (7.41) for the cylinder     p a0 R , 2c0 R (¯ x, 0) + Q 2 4 and a proper choice of A. In Lemma 7.20, take ν = ν˜ from Lemma 7.21 and determine the correν ). Then use Lemma 7.21 for R = 2, µ+ = 0, ω = 1, m1 = 0 sponding δ ∗ = δ ∗ (˜ ν ), over the cylinders and m2 such that 2−m2 = δ ∗ (˜   (0, t¯) + K2 × (−2m2 (p−2) 2p , 0) = [(0, t¯) + Q2 (0, m2 )] as long as they are contained in Q2 , that is, for t¯ satisfying −2p + 2m2 (p−2) 2p ≤ t¯ ≤ 0. Then

¯ t) > − 1 ≤ ν˜ |Q2 (0, m2 )| (x, t) ∈ [(0, t¯) + Q2 (0, m2 )] : θ(x, 2m2

for each one of the cylinders [(0, t¯) + Q2 (0, m2 )] (since (7.40) holds for all t ∈ [−2p , 0]). Therefore we conclude that ¯ t) ≤ − θ(x,

1 2m2 +1

,

142

7 Phase Transitions: The Doubly Singular Stefan Problem

a.e. (x, t) ∈ [(0, t¯) + Q1 (0, m2 )] = K1 × (t¯ − 2m2 (p−2) , t¯), for all t¯ ∈ [−2p + 2m2 (p−2) 2p , 0]. Since −2p + 2m2 (p−2) 2p − 2m2 (p−2) < −1, we get ¯ t) ≤ − 1 , a.e. (x, t) ∈ Q1 . θ(x, 2m2 +1 Returning to the initial variables and function we arrive at   ¯ − t0 ω t + x + K2c0 R ] × t¯ − , t¯ . θ(x, t) ≤ µ − m2 +s2 , a.e. (x, t) ∈ [¯ 2 4p   Since t0 ∈ t¯ − d∗ Rp , t¯ − ν20 d∗ Rp , we have  p  p   t¯ − t0 R ν0 ∗ R ∗ ¯ ¯ ¯ t− ∈ t−d ,t − d 4p 4 2 4 and then θ(x, t) ≤ µ −



ω

+

2m2 +s2

,

ν0 a.e. (x, t) ∈ Kc0 R × t¯ − d∗ 2



R 4

p

 ¯ ,t

¯ ∈ KR(ω) . since [¯ x + K2c0 R ] ⊃ Kc0 R , independently of the location of x Now we just have to make use of the arbitrary choice of t¯ in (7.16) to conclude that ω θ(x, t) ≤ µ+ − m2 +s2 , 2  p   R ν0 ,0 . a.e. (x, t) ∈ Kc0 R × −(a0 − d∗ )Rp − d∗ 2 4 Take A such that

 (p−1)(2−p) a0 A = ∗ ≥ 2. 2 d

(7.42)

Then 22p+1 − ν0 ν0 a0 ⇔ −(a0 − d∗ )Rp − d∗ ≥ d∗ 22p+1 − 1 2



R 4

p

a0 ≤− 2



R 4

p

and consequently ω θ(x, t) ≤ µ − s3 , 2 +

 a.e. (x, t) ∈ Q

a0 2



R 4

p

 , c0 R

taking s3 = m2 + s2 . An immediate consequence of Proposition 7.22 is our final result.

 

7.5 The Effect of the Singularity in the Principal Part

143

Corollary 7.23. If the second alternative holds, there exists σ1 ∈ (0, 1), depending only on the data and ω, such that ess osc

Q(

a0 2

p (R 4 ) ,c0 R)

θ ≤ σ1 ω.

(7.43)

Remark 7.24. Note that we have only imposed two conditions on A: A > 2 and (7.42). So we can take 1

A = 21+ (p−1)(2−p) > 2 and conclude that A is independent of ω. Remark 7.25. As far as B is concerned, we have B > 2s2

and

B ≥ 2s1 ,

where, recalling the choice of A, s1 satisfies s1 > log2 A + and s1 > 3 + and

p+1 p =1+ (p − 1)(2 − p) (p − 1)(2 − p)

2C (p−1)(2−p) C A = 3 + 22+(p−1)(2−p) , ν1 ν1

 

ω s2 > max 1 + Cω −α , 1 + log2 . µ+

Summarizing: 1

A = 21+ (p−1)(2−p) > 2  s1 > max 3 +

C 2+(p−1)(2−p) ,1 ν1 2

   n∗ > max Cω −α , log2 µω+ , 

B > max 2n∗ +1 , 2s1 .

+

p+1 (p−1)(2−p)

α=



2(p+1)(d+p) p

References

1. H.W. Alt and E. DiBenedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Sc. Norm. Sup. Pisa (IV) 12 (1985), 335-392. 2. S.N. Antontsev and S. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), 515-545. 3. D.G. Aronson, The Porous Medium Equation, in: Nonlinear Diffusion Problems (Montecatini Terme), pp. 1-46, Lecture Notes in Math. 1224, C.I.M.E. series, Springer, Berlin, 1986. 4. M. Bendahmane, K.H. Karlsen and J.M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. Models Methods Appl. Sci. 17 (2007), 783-804. 5. E. Bombieri, Ennio De Giorgi, Rend. Suppl. Acc. Lincei 8 (1997), 105-114. 6. M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion, SIAM J. Math. Anal. 38 (2006), 1288-1315. 7. L.A. Caffarelli and L.C. Evans, Continuity of the temperature in the two phase Stefan problem, Arch. Ration. Mech. Anal. 81 (1983), 199-220. 8. Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406. 9. Y.Z. Chen and E. DiBenedetto, On the local behaviour of solutions of singular parabolic equations, Arch. Ration. Mech. Anal. 103 (1988), 319-345. 10. E. DeGiorgi, Sulla differenziabilit´ a e l’analiticit´ a delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3 (1957), 25-43. 11. E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. 121 (1982), 131-176. 12. E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Sc. Norm. Super. Pisa 13 (1986), 487-535. 13. E. DiBenedetto, The flow of two immiscible fluids through a porous medium. Regularity of the saturation, in: “Theory and Applications of liquid crystals” (Minneapolis, 1985), 123-141, IMA Vol. Math. Appl. 5, Springer-Verlag, New York, 1987.

146

References

14. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Series Universitext, New York, 1993. 15. E. DiBenedetto, Parabolic Equations with Multiple Singularities, in: Proceedings Equadiff 9 (Brno 1997), pp. 25-48, Electronic Publishing House, Stony Brook, 1998. 16. E. DiBenedetto and R. Gariepy, On the local behavior of solutions for an ellipticparabolic equation, Arch. Ration. Mech. Anal. 97 (1987), 1-17. 17. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., to appear. 18. E. DiBenedetto, U. Gianazza and V. Vespri, Sub-potential lower bounds for non-negative solutions to certain quasi-linear degenerate parabolic differential equations, Duke Math. J., to appear. 19. E. DiBenedetto, U. Gianazza and V. Vespri, Forward, backward and elliptic harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, preprint. 20. E. DiBenedetto, U. Gianazza, M. Safonov, J.M. Urbano and V. Vespri, Harnacks Estimates: Positivity and Local Behavior of Degenerate and Singular Parabolic Equations - Special Issue, Bound. Value Probl., Hindawi Publ. Corp., 2007. 21. E. DiBenedetto, J.M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations, in: Handbook of Differential Equations, Evolutionary Equations, vol. 1, pp. 169-286, Elsevier, 2004. 22. E. DiBenedetto and V. Vespri, On the singular equation β(u)t =
u, Arch. Ration. Mech. Anal. 132 (1995), 247-309. 23. U. Gianazza, B. Stroffolini and V. Vespri, Interior and boundary continuity of the weak solution of the singular equation (β(u))t = Lu, Nonlinear Anal. 56 (2004), 157-183. 24. U. Gianazza and V. Vespri, Continuity of weak solutions of a singular parabolic equation, Adv. Differential Equations 8 (2003), 1341-1376. 25. J. Hadamard, Extension a ` l’´equation de la chaleur d’un th´eorem de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337-346. 26. E. Henriques, Regularity for the porous medium equation with variable exponent: the singular case, J. Differential Equations, to appear. 27. E. Henriques and J.M. Urbano, On the doubly singular equation γ(u)t = ∆p u, Comm. Partial Differential Equations 30 (2005), 919-955. 28. E. Henriques and J.M. Urbano, Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J. 55 (2006), 1701-1722. 29. T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math. 26 (2001) 280-301. 30. D. Hoff, A scheme for computing solutions and interface curves for a doublydegenerate parabolic equation, SIAM J. Numer. Anal. 22 (1985), 687-712. 31. A.V. Ivanov, Uniform H¨ older estimates for generalized solutions of quasilinear parabolic equations admitting a doubly degeneracy, Algebra i Analiz 3 (1991), 139-179. 32. E.F. Keller and L.A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971), 225-234. 33. S.N. Kruzkov, On the a priori estimation of solutions of linear parabolic equations and of solutions of boundary value problems for a certain class of quasilinear parabolic equations, Dokl. Akad. NAUK SSSR 138 (1961), 1005-1008; (English transl.: Soviet Math. Dokl. 2 (1961), 764-767).

References

147

34. S.N. Kruzkov, A priori estimates and certain properties of the solutions of elliptic and parabolic equations of second order, Math. Sbornik 65 (1968), 522-570; (English. transl.: Amer. Math. Soc. Transl. 2 (68) (1968), 169-220. 35. S.N. Kruzkov and S.M. Sukorjanski, Boundary value problems for systems of equations of two phase porous flow type: statement of the problems, questions of solvability, justification of approximate methods, Mat. Sb. 33 (1977), 62-80. 36. T. Kuusi, Harnack estimates for supersolutions to a nonlinear degenerate parabolic equation, Helsinki University of Technology, Institute of Mathematics, Research Report A532, 2007. 37. O.A. Ladyzhenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl. Math. Mono., Vol. 23, Providence, RI, 1968. 38. O.A. Ladyzhenskaja and N.N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. 39. J.L. Lions, Quelques M´ethodes de R´ esolution d´es Probl`emes aux Limites Nonlineaires, Dunod, Paris, 1969. 40. A. Meirmanov, The Stefan Problem, Walter de Gruyter, Berlin, 1992. 41. C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. 42. J. Moser, A new proof of DeGiorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-468. 43. J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. 44. J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954. 45. B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Math. Univ. Padova 23 (1954), 422-434. 46. M.M. Porzio and V. Vespri, H¨ older estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1993), 146-178. 47. M. Ruˇ ˙ ziˇcka, Electrorheological fluids: modelling and mathematical theory, Lect. Notes Math. 1748, Springer-Verlag, Berlin, 2000. 48. P. Sachs, Continuity of solutions of singular parabolic equations, Nonlinear Anal. 7 (1983), 387-409. 49. P. Sachs, The initial and boundary value problem for a class of degenerate parabolic equations, Comm. Partial Differential Equations 8 (1983), 693-734. 50. J.M. Urbano, A free boundary problem with convection for the p-Laplacian, Rend. Mat. Appl. 17 (1997), 1-19. 51. J.M. Urbano, Continuous solutions for a degenerate free boundary problem, Ann. Mat. Pura Appl. 178 (2000), 195-224. 52. J.M. Urbano, A singular-degenerate parabolic problem: regularity up to the Dirichlet boundary, in: Free Boundary Problems - Theory and Applications I, pp. 399-410, GAKUTO Int. Series Math. Sci. Appl. 13, 2000. 53. J.M. Urbano, H¨ older continuity of local weak solutions for parabolic equations exhibiting two degeneracies, Adv. Differential Equations 6 (2001), 327-358. 54. J.M. Urbano, Regularity for partial differential equations: from De Giorgi-NashMoser theory to intrinsic scaling, in: CIM Bull. 12 (2002), 8-14.

148

References

55. J.L. V´ azquez, The Porous Medium Equation: mathematical theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. 56. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675-710, 877. 57. W.P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982), 733-748.

Index

alternative, 25, 55, 89, 114 bootstrap argument, 3 boundedness, 12 chemotaxis, 81 cylinder rescaled, 23, 53 standard parabolic, 22 De Giorgi, 3 Bombieri on, 4 class of, 4 lemma of, 18 degeneracy, 1 accommodation of the, 22, 23 in space, 113 set of, 71 two-sided, 7, 51 DiBenedetto, 5, 6 diffusion coefficient, 53, 82, 87 equation p−Laplace, 1, 4 p−Laplacian type, 47 degenerate, 1 doubly degenerate, 48 doubly singular, 107 Euler-Lagrange, 2 porous medium, 87 porous medium type, 48 quasi-linear, 4 singular, 1 with variable exponents, 87

estimates a priori, 4, 6, 13, 108 energy, 13, 14 logarithmic, 16, 28 Schauder, 3 Stampacchia, 3 fluids electrorheological, 87 immiscible, 51 saturation of, 51 function characteristic, 26 cutoff, 14, 25 Heaviside, 108 logarithmic, 15 negative part, 14 positive part, 14 graph maximal monotone, 107 regularization, 108 with multiple jumps, 108 H¨ older continuity, 3, 4, 6, 12, 24, 35, 44, 46–48, 51, 55, 82, 110, 111 inequality Cauchy, 92 H¨ older, 74, 95, 101 Harnack, 6 oscillation, 47 Poincar´e type, 19 Young, 15, 62

150

Index

intrinsic geometry, 27, 88, 112 Harnack inequality, 6 parabolic p-distance, 46 scaling, 5, 22 time-configuration, 5 Kruzkov, 5 Ladyzhenskaya, 4, 5 model chemotaxis, 81 immiscible fluids, 51 Keller-Segel, 81 nonlinear diffusion, 82 Moser, 5, 6 Nash, 5 oscillation essential, 22 reduction of, 33, 39, 94, 114, 129 phase transition, 107 latent heat of, 108 problem Hilbert’s 19th, 2, 4 Hilbert’s 20th, 2, 3 isoperimeteric, 4 minimization, 2

regularity for minimizers, 2 regularized, 108 Stefan, 107 sequence equicontinuous, 116 geometric convergence of, 18 of nested and shrinking cylinders, 6, 21, 89 recursive, 115 singularity, 1 double, 107 in the principal part, 126 in time, 113, 114, 117 solution approximate, 109 asymptotic behavior of, 82 local weak, 12, 52, 88 of elliptic equations, 2 sub-, 131 speed of propagation finite, 82 infinite, 82 Steklov average, 12 theorem Ascoli, 108 embedding, 19 volume–filling effect, 81

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