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 U...

Author: José Miguel Urbano

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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
Kρ ×{t}

(u − k)2± ζ p dx +

≤

Kρ ×{−τ }

0

p

|∇(u − k)± ζ| dx dt

−τ

Kρ

(u − k)2± ζ p dx + C

0 −τ 0

Kρ

+p −τ

Kρ

(u − k)p± |∇ζ|p dx dt

(u − k)2± ζ p−1 ζt dx dt.

(2.6)

Proof. Let ϕ = ±(uh − k)± ζ p in (2.4) and integrate in time over (−τ, t) for t ∈ (−τ, 0). The ﬁrst term gives t 1 t (uh − k)2± t ζ p dx dθ (uh )t ϕ dx dθ = 2 −τ Kρ −τ Kρ 1 1 −→ (u − k)2± ζ p dx − (u − k)2± ζ p dx 2 Kρ ×{t} 2 Kρ ×{−τ } p t (u − k)2± ζ p−1 ζt dx dθ, − 2 −τ Kρ after integrating by parts and passing to the limit in h → 0 (using Lemma 2.2). Concerning the other term, letting ﬁrst h → 0, we obtain t |∇u|p−2 ∇u h · ∇ϕ dx dθ −τ

Kρ

−→

t −τ

≥

|∇u|p−2 ∇u · ±∇(u − k)± ζ p ± p(u − k)± ζ p−1 ∇ζ dx dθ

Kρ t

−τ

|∇(u − k)± |p ζ p dx dθ Kρ

2.3 Local Logarithmic Estimates

−p 1 ≥ 2

t

|∇(u − k)± |p−1 (u − k)± ζ p−1 |∇ζ| dx dθ

−τ t

Kρ

−τ

−C(p)

15

p

|∇(u − k)± ζ| dx dθ Kρ

t

−τ

p

Kρ

(u − k)p± |∇ζ| dx dθ,

using the inequality of Young ab ≤

1 εp p a + p bp , p pε

with the choices a = (u − k)± |∇ζ| ,

b = |∇(u − k)± ζ|

p−1

,

1

ε = [2(p − 1)] p .

and

Since t ∈ (−τ, 0) is arbitrary, we can combine both estimates to obtain (2.6). Remark 2.5. In (2.6), there is an intentional ambiguity in the way we wrote p |∇(u − k)± ζ| . The gradient can either aﬀect only (u−k)± (as follows directly from the estimates in the proof) or the product (u − k)± ζ (as the extra term can clearly be absorbed into the right hand side of the estimate).

2.3 Local Logarithmic Estimates We now introduce a logarithmic function for which we obtain further local estimates. These are the subsidiary building blocks of the theory but nevertheless play a crucial role in the proof, allowing for the expansion in time to a full cylinder Q(τ, ρ) of certain results obtained for sub-cylinders of Q(τ, ρ). Given constants a, b, c, with 0 < c < a, deﬁne the nonnegative function a ± (s) := ln ψ{a,b,c} (a + c) − (s − b)± + ⎧ a ⎪ ln if b ± c ⎪ (a+c)±(b−s) ⎨ =

⎪ ⎪ ⎩

0

if s

≤ ≥

< >

s

< >

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
Kρ ×{t}

0

+C −τ

Kρ ×{−τ }

2−p p ψ ± (u) (ψ ± ) (u) |∇ζ| dx dt.

(2.8)

Kρ

Proof. Take ϕ = 2 ψ ± (uh ) (ψ ± ) (uh ) ζ p as a testing function in (2.4) and integrate in time over (−τ, t) for t ∈ (−τ, 0). Since ζt ≡ 0,

t

−τ

(uh )t Kρ

t

2 ψ ± (uh ) (ψ ± ) (uh ) ζ p dx dθ

= −τ

Kρ

= Kρ ×{t}

2 p ψ ± (uh ) ζ dx dθ t

± 2 ψ (uh ) ζ p dx −

Kρ ×{−τ }

± 2 ψ (uh ) ζ p dx.

2.4 Some Technical Tools

17

From this, letting h → 0, t (uh )t 2 ψ ± (uh ) (ψ ± ) (uh ) ζ p dx dθ −→

−τ

Kρ

Kρ ×{t}

± 2 p ψ (u) ζ dx −

± 2 p ψ (u) ζ dx. Kρ ×{−τ }

As for the remaining term, we ﬁrst let h → 0, to obtain t p−2 |∇u| ∇u · ∇ 2 ψ ± (u) (ψ ± ) (u) ζ p dx dθ −τ t

Kρ

= −τ

± 2 p ± dx dθ |∇u| 2 1 + ψ (u) (ψ ) (u) ζ p

Kρ t

|∇u|

+p −τ

≥

t

−τ

|∇u|

p

Kρ

− 2(p − 1) ≥−C

∇u · ∇ζ

2 ψ ± (u) (ψ ± ) (u) ζ p−1 dx dθ

Kρ

p−2

t

−τ

2 2 1 + ψ ± (u) − ψ ± (u) (ψ ± ) (u) ζ p dx dθ t

p−1

−τ

2−p p ψ ± (u) (ψ ± ) (u) |∇ζ| dx dθ

Kρ

2−p p ψ ± (u) (ψ ± ) (u) |∇ζ| dx dθ.

Kρ

Since t ∈ (−τ, 0) is arbitrary, we can combine both estimates to obtain (2.8).

2.4 Some Technical Tools We gather in this section a few technical facts that, although marginal to the theory, are essential in establishing its main results. 1. A Lemma of De Giorgi Given a continuous function v : Ω → R and two real numbers k1 < k2 , we deﬁne [v > k2 ] := {x ∈ Ω : v(x) > 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 1 and α > 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
are all convex. The following theorem holds (cf. [14, page 5]). Theorem 2.12. Let v ∈ W 1,p (Ω), p ≥ 1 and assume that the set Ξ := [v = 0] ∩ [ϕ = 1] has positive measure. There exists a constant C, depending only on d and p, and independent of v and ϕ, such that p

|v| ϕ dx ≤ C Ω

|diam Ω| |Ξ|

dp

(d−1)p d

p

|∇v| ϕ dx. Ω

5. Constants With C we denote constants that depend only on d and p; these constants may be diﬀerent if they appear in diﬀerent lines.

3 The Geometric Setting and an Alternative

We go back to equation ut − div |∇u|p−2 ∇u = 0

(3.1)

and focus on the degenerate case p > 2. Results on the continuity of solutions at a point consist basically in constructing a sequence of nested and shrinking cylinders with vertex at that point, and in showing that the essential oscillation of the solution in those cylinders converges to zero as the cylinders shrink to the point. (x0 , t0 ) •

This iterative procedure is based on energy and logarithmic estimates and works well with the standard parabolic cylinders if these estimates are homogeneous. The idea goes back to the work of De Giorgi, Moser and the Russian school (cf. [10], [42] and [37]), as explained in the introduction. For degenerate or singular equations, the energy and logarithmic estimates are not homogeneous, as we have seen in the previous chapter. They involve

22

3 The Geometric Setting and an Alternative

integral norms corresponding to diﬀerent powers, namely the powers 2 and p. To go about this diﬃculty, the equation has to be analyzed in a geometry dictated by its own degenerate structure. This amounts to rescale the standard parabolic cylinders by a factor that depends on the oscillation of the solution. This procedure of intrinsic scaling, which can be seen as an accommodation of the degeneracy, allows for the restoration of the homogeneity in the energy estimates, when written over the rescaled cylinders. We can say heuristically that the equation behaves in its own geometry like the heat equation. Let us make this idea precise.

3.1 A Geometry for the Equation The standard parabolic cylinders (x0 , t0 ) + Q(R2 , R) reﬂect the natural homogeneity between the space and time variables for the heat equation. Indeed, if u(x, t) is a solution, then u(εx, ε2 t), ε ∈ R, is also a solution, i.e., the equation remains invariant through a similarity transformation of the space-time variables that leaves constant the ratio |x|2 /t. When dealing with the degenerate PDE (3.1), one might think, at ﬁrst sight, that the adequate cylinders to perform the iterative method described above were cylinders of the form Q(Rp , R), that correspond to the similarity scaling |x|p /t of the equation. But a more careful analysis shows that this is not to be expected. Indeed, it would work for the homogeneous equation (up−1 )t − div |∇u|p−2 ∇u = 0 but not for the inhomogeneous equation (3.1). By analogy, and in order to gain some hindsight on how to proceed, we recast (3.1) in the form u 2−p c

(up−1 )t − div |∇u|p−2 ∇u = 0,

for an appropriate constant c. This shows that the homogeneity can be recovered at the expense of a scaling factor, that depends on the solution itself and, modulo a constant, looks like u2−p . The following is a sophisticated and rigorous way of implementing this heuristic reasoning. Consider 0 < R < 1, suﬃciently small so that Q(R2 , R) ⊂ ΩT , and deﬁne the essential oscillation of the solution u within this cylinder ω := ess 2osc u = µ+ − µ− , Q(R ,R)

where µ+ := ess sup u Q(R2 ,R)

and

µ− := ess 2inf u. Q(R ,R)

3.1 A Geometry for the Equation

23

Then construct the rescaled cylinder Q(a0 Rp , R) = KR (0) × (−a0 Rp , 0) ,

with

a0 =

ω 2−p , 2λ

(3.2)

where λ > 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<

ω . 2λ

(3.3)

Indeed, if this does not hold, we have ω ≤ 2λ R and there is nothing to prove since the oscillation is then comparable to the radius. Now, (3.3) implies the inclusion Q(a0 Rp , R) ⊂ Q(R2 , R) and the relation ess osc

Q(a0 Rp ,R)

u≤ω

(3.4)

which will be the starting point of an iteration process that leads to the main results. The schematics below give an idea of the stretching procedure, commonly referred to as accommodation of the degeneracy (the pictures are distorted on purpose in the t-direction). (0, 0) •

(0, 0) •

R

R

Rp

Q(Rp , R) a0 Rp

Q(a0 Rp , R)

Q(R2 , R)

Q(R2 , R)

24

3 The Geometric Setting and an Alternative

Note that we had to consider the cylinder Q(R2 , R) and assume (3.3), so that (3.4) would hold for the rescaled cylinder Q(a0 Rp , 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. We now consider subcylinders of Q(a0 Rp , R) of the form (0, t∗ ) + Q(θRp , R) ,

with

θ=

ω 2−p 2

(3.5)

that are contained in Q(a0 Rp , R) provided 2p−2 − 2λ(p−2)

Rp < t∗ < 0. ω p−2

(3.6)

Once λ is chosen, we may redeﬁne it, putting λ∗ =

[p − 1] [λ] + 1 > 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

≤C

0

KRn ×{t}

(u − kn )2− ζnp dx +

2p(n+1) ≤C Rp

0

(u −

p −θRn KRn

kn )p−

p

|∇(u − kn )− ζn | dx dt

p −θRn

(u − kn )p− |∇ζn |p dx dt + p

p −θRn KRn

0

0

KRn

(u − kn )2− ζnp−1 (ζn )t dx dt

p −θRn KRn

1 dx dt + θ

0

p −θRn

(u − KRn

kn )2−

dx dt .

Next, observe that either (u − kn )− = 0 or (u − kn )− = (µ− − u) +

ω ω ω + n+2 ≤ , 4 2 2

and thus, since 2 − p < 0, p (u − kn )2− = (u − kn )2−p − (u − kn )− ω 2−p ≥ (u − kn )p− 2 = θ (u − kn )p− ,

recalling that θ = gral norms, θ sup p −θRn
≤C

ω 2−p 2

KRn ×{t}

2p(n+1) Rp

. We obtain, homogenizing the powers in the inte-

(u − kn )p− ζnp dx +

0 p −θRn

p

|∇(u − kn )− ζn | dx dt KRn

0 ω p 1 ω 2 + χ{(u−kn )− >0} dx dt, p 2 θ 2 −θRn KRn

where χE denotes the characteristic function of the set E. Finally, divide throughout by θ to get

3.2 The First Alternative

sup p −θRn
KRn ×{t}

(u − kn )p− ζnp dx +

1 θ

27

0

p

p −θRn

|∇(u − kn )− ζn | dx dt KRn

2p(n+1) ω p 1 0 ≤C χ{(u−kn )− >0} dx dt. Rp 2 θ −θRnp KRn

(3.10)

The next step, in which the intrinsic geometric framework is crucial, is to perform a change in the time variable, putting t = t/θ, and to deﬁne u(·, t) := u(·, t) ,

ζn (·, t) := ζn (·, t).

We obtain the simpliﬁed inequality 2pn ω p 0 (u − kn )− ζn p p ≤ C χ{(u−kn )− >0} dx dt, p V (Q(Rn ,Rn )) p Rp 2 −Rn KRn (3.11) which reveals the appropriate functional framework. To conclude, deﬁne, for each n, 0 χ{(u−kn )− >0} dx dt An = p −Rn

and observe that 1

ω p

2p(n+2)

2

KRn

p

An+1 = |kn − kn+1 | An+1 p

≤ (u − kn )− p,Q(Rp ,Rn+1 ) n+1 p ≤ (u − kn )− ζn p,Q(Rp ,R ) n n p p ≤ C (u − kn )− ζn V p (Q(Rp ,R )) And+p n n p 2pn ω p 1+ d+p ≤C p An . R 2

(3.12)

[The ﬁrst two inequalities follow from the deﬁnition of An and the fact that kn+1 < kn ; the third inequality is a consequence of Theorem 2.11 and the last one follows from (3.11).] Next, deﬁne the numbers Xn =

An , |Q(Rnp , Rn )|

p divide (3.12) by Q(Rn+1 , Rn+1 ) and obtain the recursive relation p 1+ d+p

Xn+1 ≤ C 4pn Xn

,

for a constant C depending only upon d and p. By Lemma 2.9 on fast geometric convergence, if X0 ≤ C −

d+p p

4−

(d+p)2 p

=: ν0

(3.13)

28

3 The Geometric Setting and an Alternative

then Xn −→ 0.

(3.14)

But (3.13) is precisely our hypothesis (3.8), for the indicated choice of ν0 , and from (3.14) we immediately obtain, returning to the original variables, ω = 0. (x, t) ∈ Q θ( R2 )p , R2 : u(x, t) ≤ µ− + 4 Remark 3.2. The constant ν0 , that appears in the formulation of the alternative, is now ﬁxed by (3.13). Note that indeed ν0 ∈ (0, 1).

3.3 The Role of the Logarithmic Estimates: Expansion in Time Our next aim is to show that the conclusion of Lemma 3.1 holds in a full cylinder Q(τ, ρ). The idea is to use the fact that at the time level p (3.15) − t := t∗ − θ R2 the function u(x, − t ) is strictly above the level µ− + ω4 in the cube K R , and 2 look at this time level as an initial condition to make the conclusion hold up to t = 0 in a smaller cylinder, as sketched in the following diagram:

− t

−

(0, 0) •

(0, 0) •

(0, t∗) •

(0, t∗) •

θRp

− t

−

θRp

As an intermediate step we need the following lemma, in which the use of the logarithmic estimates is crucial.

3.3 The Role of the Logarithmic Estimates: Expansion in Time

29

Lemma 3.3. Assume (3.8) holds for some t∗ as in (3.6) and that (3.3) is in force. Given ν∗ ∈ (0, 1), there exists s∗ ∈ N, depending only on the data, such that ω t, 0). x ∈ K R4 : u(x, t) < µ− + s∗ ≤ ν∗ K R4 , ∀t ∈ (− 2 Proof. We use the logarithmic estimate (2.8) applied to the function (u − k)− in the cylinder Q( t, R2 ), with the choices k = µ− +

ω 4

and

c=

ω 2n+2

,

where n ∈ N will be chosen later. In this cylinder, we have k−u≤

− Hu,k

ω ω − = ess sup u − µ − ≤ . 4 − 4 Q( t, R )

(3.16)

2

− If Hu,k ≤

ω 8,

− the result is trivial for the choice s∗ = 3. Assuming Hu,k >

ω 8,

recall from section 2.3 that the logarithmic function ψ − (u) is deﬁned in the whole of Q( t, R2 ) and it is given by

ψ −H − ,k, { u,k

⎧ ⎪ ⎪ ⎪ ⎨ ln ω 2n+2

}

(u) =

⎪ ⎪ ⎪ ⎩

− Hu,k − Hu,k +u−k+

ω

if u < k −

ω 2n+2

if u ≥ k −

ω 2n+2 .

2n+2

0

From (3.16), we estimate ψ − (u) ≤ n ln 2 and

since

− Hu,k − Hu,k +u−k+

≤

ω

ω 4 ω

= 2n

(3.17)

2n+2

2n+2

2−p p−2 ω p−2 − − (u) = Hu,k +u−k+c ≤ . ψ 2

(3.18)

Now observe that, as a consequence of Lemma 3.1, we have u(x, − t ) > 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
K R ×{t} 2

≤C

0

− t KR

2−p p ψ − (u) (ψ − ) (u) |∇ζ| dx dt.

(3.19)

2

The right hand side is estimated above, using (3.17) and (3.18), by ω p−2 8 p t K R ≤ C n 2λ(p−2) K R , C n(ln 2) 2 4 2 R since, by (3.15),

ω 2−p Rp . 2λ We estimate below the left hand side of (3.19) by integrating over the smaller set ω S = x ∈ K R : u(x, t) < µ− + n+2 ⊂ K R , 4 2 2 and observing that in S, ζ = 1 and t ≤ a0 Rp =

− Hu,k − Hu,k +u−k+

ω 2n+2

− is a decreasing function of Hu,k because u − k + − Hu,k − Hu,k

+u−k+

since u − µ− <

ω 2n+2

ω 2n+2

≥

ω 4

ω 4

+u−k+

ω 2n+2

=

u−

ω 2n+2

< 0. Thus, from (3.16),

ω 4 µ− +

>

ω 2n+2

ω 4 ω

= 2n−1

2n+1

in S. Therefore,

− 2 n−1 2 ψ (u) ≥ ln 2 = (n − 1)2 (ln 2)2

in S.

Combining these estimates in (3.19), we get n ω λ(p−2) R , 2 K x ∈ K R4 : u(x, t) < µ− + n+2 ≤ C 4 2 (n − 1)2 for all t ∈ (−tˆ, 0), and to prove the lemma we choose s∗ = n + 2

with

n>1+

2C λ(p−2) 2 . ν∗

3.4 Reduction of the Oscillation

31

3.4 Reduction of the Oscillation We now state the main result in the context of the ﬁrst alternative. Proposition 3.4. Assume (3.8) holds for some t∗ as in (3.6) and that (3.3) is in force. There exists s1 ∈ N, depending only on the data, such that ω R . t, u(x, 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
in KRn ,

KRn ×{t}

(u − kn )2− ζnp dx +

≤C ≤C

0

− t KRn

2p(n+4) Rp

0

n = 0, 1, . . . ,

− t KRn

p

|∇(u − kn )− ζn | dx dt

(u − kn )p− |∇ζn |p dx dt

0

− t KRn

(u − kn )p− dx dt.

From (3.15), we estimate t ≤ a0 Rp =

ω 2−p Rp , 2λ

(3.20)

32

3 The Geometric Setting and an Alternative

where a0 is deﬁned in (3.2). From this, ω 2−p (u − kn )p− 2s1 s1 p−2 2 t ≥ (u − kn )p− λ 2 Rp t ≥ R p (u − kn )p− ,

(u − kn )2− ≥

2 p p−2 .

provided s1 > λ +

Dividing now by

R p

sup

− t
t ( R2 )p

KRn ×{t}

(u −

kn )p− ζnp

≤C

2

pn

t

dx +

0

− t KRn

0

throughout (3.20) gives p

2

t

− t KRn

|∇(u − kn )− ζn | dx dt

(u − kn )p− dx dt. ( R )p

The change of the time variable t = t 2t , along with deﬁning the new function u(·, t) := u(·, t), leads to the simpliﬁed inequality p

(u − kn )− ζn V p

(Q(( R2 )p ,Rn ))

≤C

2pn ω p 0 χ{u
Deﬁne, for each n, An =

0

p −( R 2 )

KRn

χ{(u−kn )− >0} dx dt.

By a reasoning similar to the one leading to (3.12), we obtain 1 2p(n+2)

ω p p An+1 = |kn − kn+1 | An+1 2s1 p ≤ (u − kn )− p,Q ( R )p ,R ( 2 n+1 ) p ≤ (u − kn )− ζn p,Q ( R )p ,R ( 2 n) p

p

≤ C (u − kn )− ζn V p Q ( R )p ,R And+p ( ( 2 n )) p 2pn ω p 1+ d+p ≤C R p An . ( 2 ) 2s1

3.4 Reduction of the Oscillation

33

Next, deﬁne the numbers An , Xn = R p Q ( ) , Rn 2 and divide the previous inequality by Q ( R2 )p , Rn+1 to obtain the recursive relations 1+ p Xn+1 ≤ C 4pn Xn d+p . By Lemma 2.9 on fast geometric convergence, if X0 ≤ C −

d+p p

4−

(d+p)2 p

=: ν∗ ∈ (0, 1)

(3.21)

then Xn −→ 0.

(3.22)

Apply Lemma 3.3 with such a ν∗ and conclude that there exists s∗ =: s1 , depending only on the data, such that ω t, 0), x ∈ K R4 : u(x, t) < µ− + s1 ≤ ν∗ K R4 , ∀t ∈ (− 2 which is exactly (3.21). Since (3.22) implies that An → 0, we conclude that (x, t) ∈ Q R p , R : u(x, t) ≤ µ− + ω 2 s +1 1 8 2 ω R : u(x, t) ≤ µ− + s1 +1 = 0. = (x, t) ∈ Q t, 8 2

We ﬁnally reach the conclusion of the ﬁrst alternative, namely the reduction of the oscillation. Corollary 3.5. Assume (3.8) holds for some t∗ as in (3.6) and that (3.3) is in force. There exists a constant σ0 ∈ (0, 1), depending only on the data, such that ess osc

p R Q(θ( R 8 ) , 8 )

u ≤ σ0 ω.

Proof. By Proposition 3.4, there exists s1 ∈ N such that ess inf u ≥ µ− + Q( t, R 8 )

ω 2s1 +1

(3.23)

34

3 The Geometric Setting and an Alternative

and thus ess osc u = ess sup u − ess inf u Q( t, R 8 )

Q( t, R 8 )

Q( t, R 8 )

ω ≤ µ+ − µ− − s1 +1 2 1 = 1 − s1 +1 ω. 2 p p t = −t∗ + θ R2 , with t∗ < 0, we have Since θ R8 ≤ p R R R ⊂Q t, , Q θ , 8 8 8 and the corollary follows with σ0 = 1 − 2s11+1 .

4 Towards the H¨ older Continuity

Assume now that the second alternative (3.9) holds true: 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 . (4.1) |Q(θRp , R)| This is somehow the unfavorable case but we will show that a conclusion similar to (3.23) can still be reached. The idea is to exploit the fact that (4.1) holds for all cylinders of the type mentioned and add up that information to obtain the desired conclusion. Recall that the constant ν0 has already been quantitatively determined by (3.13), and it is now ﬁxed. We continue to assume that (3.3) is in force.

4.1 Expanding in Time Fix a cylinder (0, t∗ ) + Q(θRp , R) ⊂ Q(a0 Rp , R) for which (4.1) holds. Then there exists a time level ν0 t◦ ∈ t∗ − θRp , t∗ − θRp 2 such that 1 − ν0 ω |KR | . (4.2) ≤ x ∈ KR : u(x, t◦ ) > µ+ − 2 1 − ν0 /2 Otherwise, for all t ∈ t∗ − θRp , t∗ − ν20 θRp , we would have ω (x, t) ∈ (0, t∗ ) + Q(θRp , R) : u(x, 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◦
KR ×{t}

t∗

+ 2 p ψ (u) ζ dx ≤

37

+ 2 p ψ (u) ζ dx KR ×{t◦ }

2−p p ψ + (u) (ψ + ) (u) |∇ζ| dx dt.

+C t◦

(4.6)

KR

The ﬁrst integral on the right hand side can be bounded above using (4.2) and taking into account that ψ + (u) vanishes on the set {x ∈ KR : u(x, ·) ≤ k} . This gives, using also (4.4), + 2 p 1 − ν0 |KR | . ψ (u) ζ dx ≤ n2 (ln 2)2 1 − ν0 /2 KR ×{t◦ } To bound the second integral we use (4.4) and (4.5):

t∗

2−p p ψ + (u) (ψ + ) (u) |∇ζ| dx dt

C t◦

KR

ω p−2 ≤ C n(ln 2) (σR)−p (t∗ − t◦ ) |KR | 2 p ω p−2 1 θRp |KR | ≤Cn 2 σR 1 ≤ C n p |KR | , σ 2−p since t∗ − t◦ ≤ θRp and θ = ω2 . The left hand side is estimated below by integrating over the smaller set ω S = x ∈ K(1−σ)R : u(x, t) > µ+ − n+1 ⊂ KR 2 and observing that in S, ζ = 1 and + Hu,k + Hu,k −u+k+

ω 2n+1

+ ω is a decreasing function of Hu,k because −u + k + 2n+1 < 0. Thus, from (4.3), + Hu,k + Hu,k

−u+k+

ω 2n+1

since −u + µ+ <

≥

ω 2n+1

ω 2

ω 2

−u+k+

ω 2n+1

=

−u +

ω 2 µ+

+

>

ω 2n+1

in S. Therefore

+ 2 n−1 2 ψ (u) ≥ ln 2 = (n − 1)2 (ln 2)2

in S

ω 2 ω 2n

= 2n−1

38

4 Towards the H¨ older Continuity

and, from this, sup

t◦
KR ×{t}

+ 2 p ψ (u) ζ dx ≥ (n − 1)2 (ln 2)2 |S|.

Combining the three estimates, we arrive at 2 1 − ν0 n 1 n |KR | + C |S| ≤ |KR | 2 n−1 1 − ν0 /2 (n − 1) σ p 2 1 − ν0 C n |KR | + |KR | . ≤ n−1 1 − ν0 /2 nσ p On the other hand, ω x ∈ KR : u(x, t) > µ+ − n+1 2 ≤ x ∈ K(1−σ)R : u(x, t) > µ+ − ≤ |S| + dσ |KR | , and thus

ω + KR \ K(1−σ)R

2n+1

ω x ∈ KR : u(x, 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
≤C

KRn ×{t}

0

−

a0 2

p Rn

−

p Rn

KRn

(u − kn )p+ |∇ζn |p dx dt KRn

0

−

a0 2

p Rn

0

+C 2p(n+1) ≤C Rp

a0 2

−

a0 2

p Rn

KRn

(u − kn )2+ ζnp−1 (ζn )t dx dt

(u − kn )p+ dx dt KRn

2 + a0

0

−

a0 2

p Rn

(u − kn )2+ dx dt .

KRn

(4.10)

40

4 Towards the H¨ older Continuity

Observe that (u − kn )p+ ≥ (u − kn )2+ = (u − kn )2−p +

ω 2−p (u − kn )p+ . 2λ

Therefore, from (4.10), ω 2−p sup (u − kn )p+ ζnp dx a0 p 2λ KRn ×{t} − 2 Rn
a0 2

p Rn

KRn

0 2 ω 2 2p(n+1) ω p + χ{(u−kn )+ >0} dx dt. ≤C a p Rp 2λ a0 2λ − 20 Rn KRn ω 2−p and divide by a0 in the previous inequality to get Recall that a0 = λ 2 0 1 p (u − kn )p+ ζnp dx + |∇(u − kn )+ ζn | dx dt sup a0 p a0 p a 0 − 2 Rn KRn − 2 Rn 0} dx dt. ≤ p R 2λ a0 − a20 Rnp KRn Next, perform a change in the time variable, putting t t= a0 /2 and deﬁning u(·, t) = u(·, t)

and

ζn (·, t) = ζn (·, t),

to obtain the simpliﬁed inequality (u − kn )+ ζn p p V (Q(Rp ,R n

Deﬁne, for each n,

n ))

An =

≤

0

p −Rn

C2pn ω p 0 χ{(u−kn )+ >0} dx dt. p Rp 2λ −Rn KRn KRn

χ{(u−kn )+ >0} dx dt

and estimate, as before, ω p 1 p An+1 = |kn+1 − kn | An+1 2p(n+2) 2λ p ≤ (u − kn )+ p,Q(Rp ,Rn+1 ) n+1 p ≤ (u − kn )+ ζn p,Q(Rp ,R ) n n p p ≤ C (u − kn )+ ζn V p (Q(Rp ,R )) And+p n n p C2pn ω p 1+ d+p ≤ An . (4.11) p λ R 2

4.3 Deﬁning the Geometry

Deﬁne the numbers Xn =

41

An , |Q(Rnp , Rn )|

p divide (4.11) by Q(Rn+1 , Rn+1 ) , and obtain the recursive relation p 1+ d+p

Xn+1 ≤ C 4pn Xn

.

By Lemma 2.9 on fast geometric convergence, if X0 ≤ C −

d+p p

4−

(d+p)2 p

=: ν∗

(4.12)

then Xn −→ 0.

(4.13)

Thus if (4.12) holds, then p =0 (x, t) ∈ Q a0 R , R : u(x, t) > µ+ − ω 2 2 2 2λ+1 and the result follows. We are then left to prove (4.12).

4.3 Deﬁning the Geometry We now reach the crucial moment of ﬁxing λ and, consequently, determining the length of the cylinder Q(a0 Rp , R) (recall the deﬁnition of a0 ). The statement of Proposition 4.3 has then a double scope: we determine a level µ+ −

ω 2λ+1

and a cylinder (ﬁxing λ and consequently a0 ) such that the conclusion holds in that particular cylinder. Proof of Proposition 4.3(continued). To simplify the symbolism introduce the sets ω Bσ (t) = x ∈ KR : u(x, t) > 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
KRn ×{t}

(vω − kn )2+ ζn2 + ψ

ω 0 2 |ζn ∇(vω − kn )+ | 2s1 − t KRn

2 ω 22(n+4) ω 2 0 4C1 2 ψ 2s1ω−1 +8ψ s ≤ χ{vω ≥kn } . 21 R2 2s1 (2C0 − 1) ψ 2ωs1 − t KRn 2 R t , with the new function The change in the time variable t = 2 t

vω (·, t) = vω (·, t), leads to

R 2 2 ψ 2ωs1

sup

t

≤

4C1 2 ψ (2C0 −

2

2 −( R 2 )

2

ω 2s1 −1 2 1) ψ 2ωs1

|ζn ∇(vω − kn )+ | KRn

22(n+4) ω 2 0 χ{vω ≥kn } , R2 2s1 2 −( R KRn 2 )

+8

R 2

after multiplication of the whole expression by suﬃciently large, we have

γ=

ψ

2 ω 2 s1

t

. Now, if s1 is chosen

R 2 ψ

and obtain, with

(vω − kn )2+ ζn2

0

+

KRn ×{t}

2 −( R 2 )

2 ω 2 s1

t

≥1

4C1 2 ψ( 2s1ω−1 )2 +8 , (2C0 − 1) ψ( 2ωs1 )2

(5.16)

5.3 Focusing on One Degeneracy

65

the simpliﬁed inequality

(vω −

2 kn )+ ζn V 2 (Q(( R )2 ,Rn )) 2

22(n+4) ω 2 0 χ{vω ≥kn } . ≤γ R2 2s1 2 −( R KRn 2 )

Deﬁne, for each n, An =

0

2 −( R 2 )

KRn

χ{vω ≥kn } dx dt

and observe that the following estimates hold, by a reasoning similar to the one that led to (5.10): 1 22(n+2)

ω 2 2 An+1 = |kn+1 − kn | An+1 2s1 2 ≤ (vω − kn )+ 2,Q(( R )2 ,Rn+1 ) 2

2

≤ (vω − kn )+ ζn 2,Q(( R )2 ,Rn ) 2

≤ C (vω − ≤γ

2 kn )+ ζn V 2 (Q(( R )2 ,Rn )) 2

2

And+2

2 22(n+4) ω 2 1+ d+2 A . n R2 2s1

Next, deﬁne the numbers An , Xn = R 2 Q ( ) , Rn 2 divide the inequality by Q ( R2 )2 , Rn+1 and obtain the recursive relation 2 1+ d+2

Xn+1 ≤ γ 42n Xn

.

Using again Lemma 2.9 we conclude that if X0 ≤ γ −

d+2 2

4−

(d+2)2 2

=: ν1 ∈ (0, 1)

(5.17)

then Xn −→ 0.

(5.18)

Apply Lemma 5.4 with this ν1 and conclude that there exists s1 , depending only on the data, such that ω t, 0), x ∈ K R4 : v(x, t) ≥ 1 − s1 ≤ ν1 K R4 , ∀t ∈ (− 2

66

5 Immiscible Fluids and Chemotaxis

and obtain (5.17) as follows: X0 =

0

2 −( R 2 )

( R )2 = 2 t ≤

KR

χ{vω

≥1− 2ω s1

R Q ( )2 , R 2 4 0 χ{v ≥1−

( R2 )2 t

}

4

} 4 R Q ( )2 , R 2 4 ω t x ∈ K R : v(x, t) ≥ 1 − s1 4 2 ≤ ν1 . R 2 ( 2 ) K R 4 − t KR

ω 2s1

Now the conclusion easily follows from (5.18). In fact, observe that Rn

R 8

kn 1 −

and

ω , 2s1 +1

and since (5.18) implies that An → 0, we conclude that 2 R R ω : vω (x, t) ≥ 1 − s1 +1 , (x, t) ∈ Q 2 8 2 ω R : v(x, t) ≥ 1 − s1 +1 = 0 = (x, t) ∈ Q t, 8 2

and the proposition is proved.

Remark 5.6. It is very important to remark that ν1 in the proof is independent of s1 , since otherwise the reasoning would be fallacious. In fact, we have ν1 = γ −

d+2 2

4−

(d+2)2 2

and with our assumption that ψ is a power, we have 2 ψ( 2s1ω−1 ) 4C1 2 4C1 2 2p1 2 +8 , +8 = γ= 2C0 − 1 ψ( 2ωs1 ) 2C0 − 1 which clearly shows what was claimed. This also proves the independence of s1 with respect to ω, through Lemma 5.4, together with the remark that concerning the choice (5.16), we have φ 2ωm ≥ 4ψ 2ωs1 t

R 2 2 ψ 2ωs1

because

t ≤ θR2

5.4 Behaviour Near the other Degeneracy

67

and so it can be made larger than 1 if ω p2 φ 2ωm m ω = 2 ω p1 = ω p2 −p1 2p1 s1 −mp2 −2 ≥ 1. 4ψ 2s1 4 2 s1 Since p1 > p2 and ω < 1, it is enough to choose s1 ≥

2 + mp2 , p1

that clearly does not depend on ω. Corollary 5.7. There exists a constant σ0 ∈ (0, 1), depending only on the data, such that if (5.7) holds then ess osc v ≤ σ0 ω.

(5.19)

Q( t, R 8 )

Proof. We can use Proposition 5.5 to obtain s1 ∈ N such that ess sup v ≤ 1 − Q( t, R 8 )

ω 2s1 +1

and from this we get ess osc v = ess sup v− ess inf v ≤ 1 − Q( t, R 8 )

Q( t, R 8 )

Q( t, R 8 )

and the corollary follows with σ0 = 1 −

ω 2s1 +1

1 2s1 +1

−0=

.

1−

1 2s1 +1

ω

5.4 Behaviour Near the other Degeneracy Let us now suppose that (5.7) does not hold. In that case we have the complementary condition and, since 1 − ω2 ≥ ω2 , this means that for every cylinder ∗ of the type QtR , we have ∗ ∗ ω (5.20) (x, t) ∈ QtR : v(x, t) < ≤ (1 − ν0 ) QtR . 2 We are in the case for which we have to analyze the behaviour of v near 0, the other point where α(v) degenerates. We will show that also in this case a conclusion similar to (5.19) can be reached. Recall that the constant ν0 has already been determined in the previous section and is given by (5.11). ∗ Fix a cylinder QtR ⊂ Q(θR2 , R) for which (5.20) holds. There exists a time level ! 2 2 R ν R 0 t◦ ∈ t∗ − ω , t∗ − 2 ψ ω4 ψ 4

68

5 Immiscible Fluids and Chemotaxis

such that

1 − ν0 ω |KR | . < x ∈ KR : v(x, t◦ ) < 2 1 − ν20

(5.21)

In fact, if this is not true, then t∗ − ν20 Rω2 ψ( ) ω ω 4 ◦ x ∈ K : v(x, t) < x ∈ KR : v(x, t ) < ≥ dt R 2 2 2 t∗ − ψ(Rω ) 4 1 − ν0 ν0 R 2 R2 ∗ ∗ − t + ω |KR | ≥ t − 2 ψ ω4 1 − ν20 ψ 4 = (1 − ν0 ) = (1 − ν0 )

ψ

R2 ω |KR |

4 t∗ |QR |,

which contradicts our assumption. The next lemma asserts that the set where v(x) is close to its inﬁmum is small, not only at a speciﬁc time level, but for ∗ all time levels near the top of the cylinder QtR . Lemma 5.8. There exists s2 ∈ N, depending only on the data, such that ν 2 ω 0 |KR | , x ∈ KR : v(x, t) < s2 ≤ 1 − 2 2 2 for all t ∈ t∗ − ν20 ψ(Rω ) , t∗ . 4

Proof. We use a logarithmic estimate for the function (v − k)− in the cylinder KR × (t◦ , t∗ ), with the choices k=

ω 2

and

c=

ω , 2n+1

where n ∈ N will be chosen later, as parameters in the standard logarithmic function (2.7). We have ω ω − (5.22) k − v ≤ Hv,k = ess sup v − ≤ . 2 − 2 KR ×(t◦ ,t∗ ) + + If Hv,k ≤ ω4 , the result is trivial for the choice s2 = 2. Assuming Hv,k > ω4 , recall from section 2.3 that the logarithmic function ψ − (v) is deﬁned in the − + v − k + c > 0), whole domain of v, KR × (t◦ , t∗ ) (since it is obvious that Hv,k and given by ⎧ − ⎪ Hv,k ⎪ ω ⎪ if v < k − 2n+1 ⎨ ln − ω H + v − k + − n+1 v,k 2 ψ H − ,k, ω (v) = { v,k 2n+1 } ⎪ ⎪ ⎪ ⎩ ω 0 if v ≥ k − 2n+1 .

5.4 Behaviour Near the other Degeneracy

69

From (5.22), we can easily estimate ψ − (v) ≤ n ln 2

since

− Hv,k − Hv,k +v−k+

≤

ω 2n+1

ω 2 ω

= 2n

2n+1

and the derivative (here in the nonvanishing case v < k − c) 1 2n+1 −1 − = . (v) = − ψ ≤ Hv,k + v − k + c c ω Choose a piecewise smooth cutoﬀ function 0 < ζ(x) ≤ 1, deﬁned on KR and such that, for a certain σ ∈ (0, 1), ζ = 1 in K(1−σ)R

and

|∇ζ| ≤

C σR ,

and multiply (5.2) by 2ψ − (vh ) (ψ − ) (vh ) ζ 2 . Integrating in time in (t◦ , t), with t ∈ (t◦ , t∗ ), performing the usual integrations by parts, and passing 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 . KR ×{t◦ }

KR ×{t}

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

= t◦

KR

t

α(v)∇v · ∇ζ 2ψ − (v) ψ − (v) ζ = (∗).

+2 t◦

KR

Using Young’s inequality to estimate the second term as in the proof of Lemma 5.4, we get t t − 2 2 2 α(v)|∇v| ζ (ψ ) (v) − 2 α(v)|∇ζ|2 ψ − (v) (∗) ≥ 2 t◦

≥ −2n ln 2

C σ 2 R2

t◦

KR

t t◦

KR

α(v)χ{v< ω2 } ≥ −2n ln 2

because in the relevant set α(v) ≤ C1 φ(v) ≤ C1 φ

KR

ω C ∗ ◦ , (t −t ) |K | C φ R 1 σ 2 R2 2

ω 2

.

70

5 Immiscible Fluids and Chemotaxis

Now we observe that t∗ − t◦ ≤ t∗ − t∗ +

ψ

R2 ω = 4

ψ

R2 ω, 4

and so we can condensate the two estimates in the logarithmic estimate − 2 2 − 2 2 1 φ ω2 ψ (v) ζ ≤ ψ (v) ζ + C n 2 ω |KR | sup σ ψ 4 t◦ ≤t≤t∗ KR ×{t} KR ×{t◦ } 1 − ν0 1 φ ω2 2 |KR | + C n 2 ω |KR | , ≤Cn 1 − ν20 σ ψ 4 because ψ − (v) = 0 in the set {v > 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 2l

and

Al =

0

− θ2 R2

|Al (t)| dt,

and using Lemma 2.7, obtain C Rd+1 ω ω − l+1 |Al+1 (t)| ≤ 2 |∇w|. ν0 2l 2 |KR | Al (t)\Al+1 (t) 2

Integrate this inequality in time over [− θ2 R2 , 0], use H¨older’s inequality and square both sides, to get

5.4 Behaviour Near the other Degeneracy

75

2

0 R2 (A − A ) |∇w|2 l l+1 ν04 − θ2 R2 Al (t)\Al+1 (t) 0 R2 2 ≤ C 4 (Al − Al+1 ) |∇(w − k)− | ν0 − θ2 R2 KR 2

C 2l θ 2 ω ω ω R , R φ m ≤ 4 (Al − Al+1 ) 1 + B Q l l ν0 ω 2 2 2 2 ω 2l

A2l+1 ≤ C

using also inequality (5.24). Adding these inequalities for l = s4 , s4 + 1, . . . , s5 − 1, where s5 is to be chosen, we get s" 5 −1

A2l+1

l=s4

C 2s5 ω ω B φ m ≤ 4 (As4 − As5 ) 1 + ν0 ω 2s4 2

Q θ R2 , R , 2

and since As4 − As5 ≤ Q( θ2 R2 , R) and s" 5 −1

A2l+1 ≥ (s5 − s4 )A2s5 ,

l=s4

we ﬁnally conclude that As5

1 C ≤ 2 (s5 − s4 )− 2 ν0

2s5 B 1+ ω

ω 2s4

ω 12 φ m 2

Q θ R2 , R . 2

To prove the result, we choose s5 so large that 1 λ0 C (s5 − s4 )− 2 ≤ √ ν02 2

and m0 so large that 2s5 B ω

ω 2s4

φ

ω ≤ 1. 2m0

(5.25)

Concerning the dependence on ω, we immediately conclude that, since ν0 and s4 are independent of ω (Lemma 5.3 and Corollary 5.11), so is s5 . The independence of m0 on ω is more delicate and again crucially related to the fact that φ is a power. Observe that as a consequence of this fact 1

B(s) = {(p2 + 1)s} p2 +1

and

ω=

ω p2 +1 . p2 + 1

76

5 Immiscible Fluids and Chemotaxis

So, from (5.25), we must choose m0 such that s4 ω p2 ω p2 +1 2 p2 +1 ≤ 2m0 (p2 + 1)2s5 ω

that is

s4

2m0 p2 ≥ (p2 + 1) 2s5 − p2 +1 ,

and now it is clear that m0 can be chosen independently of ω.

We can now obtain Proposition 5.10 by rephrasing the contents of Proposition 5.12. In fact, w(x, t) <

ω ω ⇐⇒ v(x, t) < s5 2s5 p2 +1 2

5 and we conclude putting s3 = p2s+1 . We next explain how to choose the constant m (independently of ω) and consequently ﬁx the height of the cylinder Q(θR2 , R). We follow closely the idea in [13]. Let

φ1 = φ

ω , 2s3

φ2 = φ

ω , 2s3 +2

µ = φ2

φ1 φ2

d+2 2 ,

and choose m > m0 as the smallest real number such that φ

µ ω = n0 ,

for some integer n0 ∈ N.

(5.26)

2m

Since φ is a power, it is clear that m is independent of ω. Then break Q(θR2 , R) again, this time into n0 subcylinders of the form R2 R2 , −(j − 1) ; j = 1, 2, . . . , n0 . QjR = KR × −j µ µ Since these cylinders are disjoint and they exhaust Q(θR2 , R), from Proposition 5.10 it follows that ω (5.27) ∃j0 ∈ {1, . . . , n0 } : (x, t) ∈ QjR0 : v(x, t) < s3 ≤ λ0 QjR0 . 2 We now use this information to show that in a smaller cylinder the function v is strictly away from the degeneracy at 0. Lemma 5.13. The number s3 can be chosen such that v(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˜
KRn ×{t}

(vω − kn )2+ ζn2 + φ

ω 0 2 |ζn ∇(vω − kn )+ | 2s6 −t˜ KRn

ω 22(n+4) ω 2 0 χ{vω ≤kn } . 2s6 R2 2s6 +1 −t˜ KRn 2 R t , with the new function The change in the time variable t = 2 t˜ ≤Cφ

vω (·, t) = vω (·, t), leads to φ

R 2 2 ω 2 s6

t˜

sup 2 −( R 2 )
KRn ×{t}

(vω − kn )2− ζn2

0

2

|ζn ∇(vω − kn )− |

+ 2 −( R 2 )

KRn

22(n+4) ω 2 0 ≤C χ{vω ≤kn } , R2 2s6 +1 2 −( R KRn 2 ) R 2 2 . Now, if s6 is chosen after multiplication of the whole expression by φ 2ωs6 t˜ suﬃciently large, we have R 2 2 ω ≥1 φ s6 t˜ 2

since φ

R 2 2 ω 2 s6

t˜

=

j0 φ

µ

ω 2 s6

≥

n0 φ

µ

ω 2 s6

=

φ 2ωm ω , φ 2 s6

which shows also that it suﬃces to pick s6 > m. The rest of the proof follows the same lines of the proof of Proposition 5.5, that consist basically in using the iteration technique and then Lemma 5.15. We omit the details. Corollary 5.17. There exists a constant σ1 ∈ (0, 1), depending only on the data, such that ess osc v ≤ σ1 ω. Q(t˜, R 8 ) Proof. It is similar to the proof of Corollary 5.7. We ﬁnd σ1 = 1 −

1 2 s6

.

An immediate consequence of Corollaries 5.7 and 5.17 is our ﬁnal result.

5.5 A Problem in Chemotaxis

81

Proposition 5.18. There exists a constant σ ∈ (0, 1), that depends only on the data, and a cylinder Q t♦ , R8 , such that ess osc v ≤ σ ω.

Q(t♦ , R 8 )

Proof. Since one of (5.7) or (5.20) has to be true, the conclusion of at least one of Corollaries 5.7 or 5.17 holds. Choosing σ = max {σ0 , σ1 } , and

t , t˜ , t♦ = min

we obtain the conclusion.

The proof of Theorem 5.2 now follows from Proposition 5.18 as in Section 4.4. We stress that the H¨ older continuity is obtained since σ in Proposition 5.18 is independent of the oscillation ω.

5.5 A Problem in Chemotaxis We now consider a similar problem ⎧ ⎨ vt − div (α(v)∇v) = −div (χvf (v)∇u) ⎩

in ut − ∆u = g(v, u)

ΩT

(5.28)

arising from the modelling of chemotaxis, a property of certain living organisms to be repelled or attracted to chemical substances. Here v = v(x, t) represents the density of a cell-population, u = u(x, t) represents the chemoattractant (repellent) concentration, α(v) is a density-dependent diﬀusion coefﬁcient and f (v(x, t)) measures the probability that a cell in position x at time t ﬁnds space in its neighboring location. The cells are attracted by the chemical and χ denotes their chemotactic sensitivity. The function g(v, u) describes the rates of production and degradation of the chemoattractant. The above model is a special case of the original Keller-Segel model [32], introduced to describe the aggregation of the cellular slime mold Dictyostelium discoideum due to the eﬀect of the cyclic Adenosine Monophosphate (cAMP), an attractive chemical signal for the amoebae. We consider the following assumptions, corresponding to the two the main features of the model: • There is a maximal density of cells, the threshold vm , such that f (vm ) = 0. Intuitively, this amounts to a switch to repulsion at high densities, sometimes referred to as volume–ﬁlling eﬀect or prevention of overcrowding (see [29]). This threshold condition has a clear biological interpretation:

82

5 Immiscible Fluids and Chemotaxis

the cells stop to accumulate at a given point of Ω after their density attains certain threshold values and the chemotactic cross diﬀusion h(v) = χvf (v) vanishes identically when v ≥ vm . • The density-dependent diﬀusion coeﬃcient α(v) degenerates for v = 0 and v = vm . This means, in particular, that there is no diﬀusion when v approaches values close to the threshold. This interpretation was proposed in [6], where the diﬀusion coeﬃcient takes the form α(v) = v(1 − v), for > 0. The main advantages of the nonlinear diﬀusion model seem to be related to the ﬁnite speed of propagation (which is more realistic in biological applications than inﬁnite speed) and the asymptotic behavior of solutions. Deﬁning new variables through v˜ =

v , vm

u ˜ = u,

we have v˜m = 1. After performing this linear transformation, we omit the tildes in the notation. A typical example of f in this case is f (v) = 1 − v. Under suitable assumptions on the data, an existence theorem is proved in [4], using a Schauder ﬁxed-point argument on a regularized problem and the compactness method. Here, we comment on the local H¨older continuity of a weak solution v of the system under the same assumptions (A1)–(A3) on α used for the immiscible ﬂuids model. The novelty with respect to the previous sections is the additional lower-order term div(χvf (v)∇u); we show that it satisﬁes the appropriate growth conditions due to its special form and the available regularity for u, which follows from the classical theory of parabolic PDEs. The pertinent deﬁnition of local weak solution can be cast in the following formulation: for every compact K ⊂ Ω and for every 0 < t < T − h, {(vh )t ϕ + (α(v)∇v)h · ∇ϕ − χ (vf (v))h ∇u · ∇ϕ} dx = 0, (5.29) K×{t}

for all ϕ ∈ H01 (K). Here, u is treated as a given function in its existence class so all the terms in the above expression have a meaning. To study the locally regularity of v we proceed exactly as before, deﬁning the same intrinsic geometry and establishing the same type of alternative. To illustrate this fact, we now prove the equivalent of Lemma 5.3. Recall that we denote ω . vω := min v, 1 − 4

5.5 A Problem in Chemotaxis

83

Taking the cylinder for which the ﬁrst alternative (5.7) holds, we deﬁne Rn =

R R + n+1 , 2 2

n = 0, 1, . . . ,

and construct the family of nested and shrinking cylinders ∗ Rn2 ∗ , t . QtRn = KRn × t∗ − ψ( ω4 ) Next, we consider piecewise smooth cutoﬀ functions 0 ≤ ξn ≤ 1, deﬁned in these cylinders, and satisfying assumptions (5.8). Letting kn = 1 −

ω ω − n+2 , 4 2

n = 0, 1, . . .

we choose as test function in (5.29) ϕ = [(vω )h − kn ]+ ξn2 and integrate in R2

time over (t∗ − ψ( ωn ) , t) for t ∈ (t∗ − 4 notation, we put

2 Rn , t∗ ) ψ( ω 4)

τn := t∗ −

with K = KRn . To simplify the

Rn2 , ψ( ω4 )

and again omit, from here on, dx and dt in all integrals. We only consider the lower order term since it encompasses the main novelty with respect to the previous sections. After passing to the limit in h, using the convergence properties of the Steklov average and Young’s inequality, we obtain t

vf (v)∇u · ξn2 ∇(vω − kn )+ + 2(vω − kn )+ ξn ∇ξn χ τn

KRn

t 1 1 ω t 2 2 M |ξn ∇(vω − kn )+ | + |∇u|2 χ{vω ≥kn } ≤ ψ 2 4 2ψ ω4 τn KRn τn KRn t ω χ{vω ≥kn } +2M |∇u||∇ξn | 4 τn KRn since (vω − kn )+ ≤ ω4 , and deﬁning M := χvf (v) L∞ (ΩT ) . Using again Young’s inequality, we bound from above by M 2 + 2M t 1 ω t 2 ψ |ξn ∇(vω − kn )+ | + |∇u|2 χ{vω ≥kn } 2 4 2ψ ω4 τn KRn τn KRn +M

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

84

5 Immiscible Fluids and Chemotaxis

We conclude with the estimate

t

|∇u| χ{vω ≥kn } ≤ 2

τn

KRn

∇u 2Lp (ΩT )

t

τn

1− p2 + Akn ,Rn (σ) dσ

where A+ kn ,Rn (σ) := {x ∈ KRn : v(x, σ) > kn } and p is chosen suﬃciently large. This is possible since, from standard parabolic theory, for all q > 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
t a0

and deﬁne the new functions ξ¯n (x, z) = ξn (x, a0 z).

u ¯ω (x, z) = uω (x, a0 z) ;

Then the above estimates read 2 ¯2 (¯ uω − kn )− ξn + sup 2
0

2 −Rn

KRn ×{z}

|∇(¯ uω − kn )− |2 ξ¯n KRn

ω 2 22(n+2) 0 χ[¯uω ≤kn ] 4 R2 2 −Rn KRn C(M, γ + ) ω 2 22(n+2) 0 + χ[¯uω ≤kn ] + 4 R2 2 ω 1+γ −Rn KRn C(M, γ + ) ω 2 22(n+2) ≤ An , 4 R2 ω 1+γ + χ[¯uω ≤kn ] . These estimates imply the inequality ≤6

for An :=

0

2 −Rn

KRn

2

(¯ uω − kn )− V 2 (Q(R2

n+1 ,Rn+1 ))

≤

C(M, γ + ) ω 2 22(n+2) An . 4 R2 ω 1+γ +

2

94

6 Flows in Porous Media: The Variable Exponent Case

Using Theorem 2.11, we get ω 2 4

1

An+1 = (kn − kn+1 )2 An+1 ≤ 22(n+1)

2 Q(Rn+1 ,Rn+1 )

(¯ uω − kn )2−

2 2 2 ≤ C(d) [¯ uω ≤ kn ] ∩ Q(Rn+1 , Rn+1 ) d+2 (¯ u − kn )− V 2 (Q(R2

n+1 ,Rn+1 ))

≤ C(d)An

2 d+2

(¯ u−

2 kn )− V 2 (Q(R2 ,Rn+1 )) n+1

,

and consequently An+1 ≤ Deﬁning Yn :=

An 2 ,R )| |Q(Rn n

2 C(M, d, γ + ) 24n 1+ d+2 An . + 2 1+γ R ω

and noting that 2

|Q(Rn2 , Rn )|1+ d+2 ≤ 2d+4 R2 , 2 |Q(Rn+1 , Rn+1 )| we arrive at the algebraic inequality Yn+1 ≤

2 C(M, d, γ + ) 4n 1+ d+2 2 Yn . + 1+γ ω

Now, by Lemma 2.9, if Y0 ≤ C(M, d, γ + )−

d+2 2

2−(d+2) ω (1+γ

d+2 2

2−(d+2) ω (1+γ

2

+ d+2 ) 2

then Yn → 0 as n → ∞. Taking ν0 := C(M, d, γ + )−

2

+ d+2 ) 2

(6.10)

the above inequality is valid since it is no other than our hypothesis. Since Rn

R , 2

kn

ω , 4

and Yn → 0 as n → ∞ implies that An → 0 as n → ∞, we obtain 2 R ω R :u ¯ω (x, z) ≤ = 0, , (x, z) ∈ Q 2 2 4 that is, ω , u(x, t) > 4

a.e. (x, t) ∈ Q a0

R 2

2

R , 2

.

The reduction of the oscillation of u follows at once.

6.3 Analysis of the Alternative

95

Corollary 6.6. There exist constants ν0 ∈ (0, 1), depending on the data and ω, and σ0 ∈ (0, 1), such that if (6.6) holds then ess osc 2 R Q a0 ( R 2 ) , 2

u ≤ σ0 ω.

(6.11)

Proof. From Proposition 6.5 we know that ess inf 2 R Q a0 ( R 2 ) , 2

Thereby, since µ+ ≥

ess sup

2 R Q a0 ( R 2 ) , 2

ω . 4

u(x, t) ≥

u,

ω + = ess osc u≤µ − 4 R 2 R Q a0 ( 2 ) , 2

1−

1 4

ω = σ0 ω.

Remark 6.7. We comment here on how to extend the proof to the case of a ﬁnite p > max{2, d} in assumption (A1). The novelty is on how to bound the term t ∗ |∇γ|2 χ[uω ≤kn ] I := 2 −a0 Rn

KRn

now that |∇γ| is not bounded. We use H¨ older’s inequality with q = p/2 to get 1− p2 I∗ ≤ M 2

t

2 −a0 Rn

KRn

χ[uω ≤kn ]

,

with M = γ L∞ (0,T ;W 1,p (Ω)) . An expression of this form is treated as in the case of a degenerate PDE with lower-order terms, for the choices q=

2p(1 + κ) ; p−2

r = 2(1 + κ);

κ=

2(p − d) dp

in the notation of [14, page 24].

6.3 Analysis of the Alternative Now we assume that (6.6) does not hold; therefore, (6.7) is in force. We will show that, in this case, we can get a result similar to (6.11). Remember that ν0 was already determined and is given by (6.10). Lemma 6.8. Assume that (6.7) holds. There exists a time level ν0 t0 ∈ −a0 R2 , − a0 R2 2

(6.12)

96

6 Flows in Porous Media: The Variable Exponent Case

such that

1 − ν0 ω + |KR | . x ∈ KR : u(x, t0 ) > µ − < 2 1 − ν20

(6.13)

Proof. If not, then ω (x, t) ∈ Q a0 R2 , R : u(x, t) > 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
, for s ≥ s1 , ν 2 ν 0 0 |As (t)| ≤ |As1 (t)| < 1 − |KR | , ∀t ∈ − a0 R2 , 0 , 2 2 by virtue of (6.15). Then, for every t ∈ − ν20 a0 R2 , 0 , ω 2s

≥ µ+ −

ω 2 s1

ω C(d) |A (t)| ≤ R |∇u| s+1 2s+1 ν0 2 [k1
12

|∇(u − k)+ |

2

Q(

ν0 2

a0 R2 ,R)

1

|As \ As+1 | 2 .

According to the previous energy estimates we get, for s = s1 , . . . , s2 − 1, A2s+1 ≤

C(M, d, γ + ) ν0 2 a R , R Q |As \ As+1 | , 0 5 + 2 ω 2 (1+γ )(d+2)

and we then add these inequalities for s = s1 , . . . , s2 − 1. ω ≤ µ+ − 2ωs2 , As+1 ≥ As2 , and therefore Since µ+ − 2s+1 s" 2 −1

A2s+1 ≥ (s2 − s1 )A2s2 .

s=s1

Note also that

s" 2 −1 s=s1

arrive at A2s2 ≤

ν 0 a0 R2 , R . Collecting results, we |As \ As+1 | ≤ Q 2 ν 2 0 2 a R , R Q 0 5 + 2 ω 2 (1+γ )(d+2) (s2 − s1 ) C(M, d, γ + )

102

6 Flows in Porous Media: The Variable Exponent Case

and the proof is complete if we choose s1 < s2 ∈ N suﬃciently large so that C(M, d, γ + ) ω

5 + 2 (1+γ )(d+2)

(s2 − s1 )

≤ ν2.

Lemma 6.13. The number ν ∈ (0, 1) can be chosen (and consequently, so can s2 ) such that 2 R ω ν0 R + a0 . , u(x, t) ≤ µ − s2 +1 , a.e. (x, t) ∈ Q 2 2 2 2

Proof. Deﬁne two sequences of positive real numbers Rn =

R R + n+1 , 2 2

kn = µ+ −

ω 2s2 +1

−

ω 2s2 +1+n

,

n = 0, 1, 2, . . .

and construct the family of nested and shrinking cylinders ν 0 a0 Rn2 , Rn . Qn = Q 2

Consider the function uω = min u, µ+ − 2s2ω+1 and, in the weak formulation (6.2), take φ = (uω − kn )+ ξn2 , where 0 ≤ ξn ≤ 1 are smooth cutoﬀ functions deﬁned in Qn and verifying ⎧ ⎪ ⎨ ξn ≡ 1 in Qn+1 ; ξn ≡ 0 on ∂p Qn ; ⎪ ⎩ |∇ξn | ≤

2n+2 R ;

|∆ξn | ≤

22(n+2) R2 ;

0 < (ξn )t ≤

22(n+2) ν0 2. 2 a0 R

Then, for t ∈ − ν20 a0 Rn2 , 0 , we have formally (again, to be rigorous we would have to argue with the Steklov averages and pass to the limit in h) 0=

−

ν0 2

ν − 20

t

+

2 a0 Rn

t 2 a0 Rn

ut (uω − kn )+ ξn2

KRn

uγ(x,t) ∇u · ∇ (uω − kn )+ ξn2 := I1 + I2 .

KRn

Arguing as in the proof of Proposition 6.5 (with the obvious changes), we obtain ω 2 22(n+2) t 1 2 2 (uω − kn )+ ξn − 3 s2 +1 χ[uω >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
ω 2 22(n+2) 6 C(M ) ≤ s2 +1 + χ[uω ≥kn ] . ν0 2 R2 ω Qn 2 a0 Introducing the change of variables z=

t ν0 2 a0

and deﬁning the new functions ν 0 u ¯ω (x, z) = uω x, a0 z , 2

ν 0 ξ¯n (x, z) = ξn x, a0 z , 2

the previous estimate reads ν0 2 ¯2 (¯ uω − kn )+ ξn + |∇(¯ uω − kn )+ |2 ξ¯n2 sup 2
(¯ uω − kn )+ V 2 (Q(R2

n+1 ,Rn+1 ))

2 ν0

≥ 1, we arrive at

ω 2 22(n+2) 1 a0 An + 2s2 +1 R2 ν0 ω C(M, d, γ + ) ω 2 22(n+2) ≤ An , d+2 + R2 ω (1+γ )( 2 ) 2s2 +1 ≤ C(M )

104

6 Flows in Porous Media: The Variable Exponent Case

where An is deﬁned as An :=

0

2 −Rn

KRn

χ[¯uω ≥kn ] .

Now, we ﬁrst obtain An+1 ≤ and then, deﬁning Yn :=

2 C(M, d, γ + ) 24n 1+ d+2 An , d+2 + 2 ω (1+γ )( 2 ) R

An 2 ,R )| , |Q(Rn n

Yn+1 ≤

we get the algebraic inequality

C(M, d, γ + ) d+2 + ω (1+γ )( 2 )

2 1+ d+2

24n Yn

.

As before, the result is proved if we can assure that Y0 ≤ C(M, d, γ + )−

d+2 2

ω (1+γ

= C(M, d, γ + )ω (1+γ

2 + (d+2) ) 4

2 + (d+2) ) 4

2−(d+2)

2

=: ν.

By Lemma 6.12, for this value of ν there exists s1 < s2 ∈ N such that (x, z) ∈ Q R2 , R : u ¯(x, z) > µ+ − 2ωs2 ≤ν |Q (R2 , R)| which implies Y0 ≤ ν. Then we can conclude that Yn → 0 when n → ∞, and the result follows. Proposition 6.14. There exist positive numbers ν0 , σ1 ∈ (0, 1), depending on the data and on ω, such that, if (6.7) holds true then ess osc2 ν R Q 20 a0 ( R 2 ) , 2

u ≤ σ1 ω.

(6.16)

Proof. The proof is trivial and similar to the proof of Corollary 6.6. We have σ1 = 1 − 2s21+1 . Proof of Proposition 6.3. Recalling the conclusions of Corollary 6.6 and Proposition 6.14, we take σ = max {σ0 , σ1 } = σ1 ,

6.3 Analysis of the Alternative

since σ0 = 1 −

1 4

<1− Q

1 2s2 +1

ν0 a0 2

= σ1 , because s2 > 1. As ν0 ∈ (0, 1),

R 2

2

R , 2

⊂ Q a0

R 2

2

R , 2

and the result follows. Remark 6.15. The extension to the singular case −1 < γ − ≤ γ(x, t) ≤ γ + < 0 is treated in [26].

105

7 Phase Transitions: The Doubly Singular Stefan Problem

Another interesting equation is the doubly singular PDE γ(u)t − div |∇u|p−2 ∇u = 0,

1 < p < 2,

where γ is a coercive, maximal monotone graph in R × R given by ⎧ s if s < 0 ⎪ ⎪ ⎨ γ(s) = [0, λ] if s = 0 ⎪ ⎪ ⎩ s + λ if s > 0,

(7.1)

(7.2)

with λ > 0. The equation is to be interpreted in the sense of the graphs, i.e., for a choice v ∈ γ(u), and exhibits a double singularity: as 1 < p < 2, its modulus of ellipticity |∇u|p−2 blows up at points where |∇u| = 0; an additional singularity occurs in the time derivative since, loosely speaking, γ (0) = ∞. Graphs γ(·) such as this one, with a single jump at the origin, arise from the weak formulation of the classical Stefan problem (cf. [40]), that corresponds to the case p = 2 and models a solid–liquid phase transition (such as water–ice) at constant temperature for a substance obeying Fourier’s law. A natural question to ask is whether the transition of phase occurs with a continuous temperature across the water–ice interface. This issue, raised initially by Oleinik in the 1950’s, and reported in the book [37], is at the origin of the modern theory of local regularity for solutions of degenerate and/or singular evolution equations. The coercivity of γ(·) is essential for a solution to be continuous, as illustrated by examples and counterexamples in [16]. Consider the more general case γ(u)t − div a(x, t, u, ∇u) = b(x, t, u, ∇u)

in

D (ΩT ),

(7.3)

with a and b satisfying the usual structure assumptions for p = 2. Solutions of (7.3) are continuous with a modulus of continuity that is not H¨ older; this was

108

7 Phase Transitions: The Doubly Singular Stefan Problem

established in [7] (for the Laplacian), [11], [48], [49], [57], for γ(·) exhibiting a single jump. This raises naturally the question of a graph γ(·) exhibiting multiple jumps and/or singularities of other nature. For these rather general graphs, in the mid 1990’s, it was established in [22] that solutions of (7.3) are continuous provided d = 2. For dimension d ≥ 3 the same conclusion holds provided the principal part of the diﬀerential equation is exactly the Laplacian. Several recent investigations have extended and improved these results for speciﬁc graphs ([23], [24]). However, for d ≥ 3, it is still an open question whether solutions of (7.3), with its full quasilinear structure (even with p = 2) and for a general coercive maximal monotone graph γ(·), are continuous in their domain of deﬁnition.

7.1 Regularization of the Maximal Monotone Graph Here, we restrict our attention to (7.1). The analysis we will perform involves regularizing the maximal monotone graph γ and obtaining a priori estimates for the regularized problem that are independent of the regularization. The ultimate goal is to use Ascoli’s theorem to obtain the continuity of the solution of the original problem by proving it is the limit of a sequence of equibounded and equicontinuous approximate solutions. We will not be concerned with problems of existence of the weak solution for boundary value problems associated with (7.1) or the convergence of the sequence of approximate solutions to the weak solution; this problem was treated in [50] (see also the classical reference [39]). The consequence of estimating uniformly the regularization of the maximal monotone graph is the appearance of a third power (power 1) in the energy estimates. We will thus be dealing with three powers (1, p and 2) related by 1 < p < 2. In this case, the price to be paid for the recovering of the homogeneity in the energy estimates is a dependence on the oscillation in the various constants that are determined along the proof. Owing to this fact we will no longer be able to exhibit a modulus of continuity but only to deﬁne it implicitly independently of the regularization. This is enough to obtain a continuous solution for the original problem, via Ascoli’s theorem, but the H¨older continuity, that holds in the case γ(s) = s, is lost. To be precise, consider the maximal monotone graph H associated with the Heaviside function ⎧ ⎨ 0 if s < 0 H(s) = [0, 1] if s = 0 ⎩ 1 if s > 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<

1 ≤ β (s) ≤ 1, 1 + λL

s ∈ R.

(7.5)

Deﬁnition 7.1. An approximate solution of equation (7.1) is a function θ ∈ L∞ (ΩT ) ∩ H 1 0, T ; L2 (Ω) ∩ L∞ 0, T ; W01,p (Ω) such that, for a.e. t ∈ (0, T ),

[γ (θ )]t ϕ + |∇θ |p−2 ∇θ · ∇ϕ dx = 0,

(7.6)

Ω×{t}

for all ϕ ∈ W01,p (Ω). We assume that the approximate solutions satisfy the uniform bound

θ L∞ (ΩT ) ≤ M. It is easily seen that (7.6) corresponds, in the sense of distributions, to the equation ut − div |∇β (u)|p−2 ∇β (u) = 0, with u = γ (θ ), which is of the form (4.19) and satisﬁes the structure conditions

110

7 Phase Transitions: The Doubly Singular Stefan Problem

|∇β (u)|p−2 ∇β (u) · ∇u ≥ and

1 1 + λL

p−1 |∇u|p

|∇β (u)|p−2 ∇β (u) ≤ |∇u|p−1 .

As a consequence of this fact, using the general theory, the approximate solutions are H¨ older continuous. But the constant p−1 1 C0 ( ) := 1 + λL deteriorates as → 0 and whatever is proved using this approach is lost in the limit. What we need to do is to obtain energy estimates that are independent of the approximating parameter . This will allow us to show that the approximate solutions are continuous independently of the approximation.

7.2 A Third Power in the Energy Estimates Consider a cylinder Q(τ, ρ) ⊂ ΩT , and a piecewise smooth cutoﬀ function ζ in Q(τ, ρ) such that 0 ≤ ζ ≤ 1;

|∇ζ| < ∞;

ζ = 0, x ∈ / Kρ .

Let k < M and ϕ = −(θ − k)− ζ p in (7.6). Integrating in time over (−τ, t) for t ∈ (−τ, 0), the ﬁrst term gives −

t

−τ

Kρ t

= −τ

Kρ

[γ (θ )]t (θ − k)− ζ p

(θ −k)−

∂t (θ −k)−

=

Kρ ×{−τ }

−p 1 ≥ 2

t

−τ

γ (k

− s)s ds ζ p

0

−

− s)s ds ζ p

0

Kρ ×{t}

γ (k

Kρ

Kρ ×{t}

γ (k − s)s ds ζ p

0 (θ −k)−

γ (k

− s)s ds ζ p−1 ζt

0

(θ −

−2p(M + λ)

(θ −k)−

k)2− ζ p t

−τ

− 2(M + λ)

Kρ ×{−τ }

(θ − k)− ζ p−1 ζt , Kρ

(θ − k)− ζ p (7.7)

7.2 A Third Power in the Energy Estimates

111

since we have, recalling (7.4),

(θ −k)−

γ (k

0

and

(θ −k)−

− s)s ds ≥

s ds = 0

(θ −k)−

γ (k

(θ −k)−

− s)s ds ≤ (θ − k)−

0

1 (θ − k)2− 2 γ (k − s) ds

0

= (θ − k)− [γ (k) − γ (θ )] ≤ 2(M + λ) (θ − k)− . Concerning the other term, we have t p−2 |∇θ | ∇θ · ∇ [−(θ − k)− ζ p ] = −τ

Kρ

−p

≥

1 2

−τ

t

−τ

t

|∇θ |

p−2

Kρ

p

|∇(θ − k)− ζ| − C(p)

−τ

t

Kρ

t

−τ

|∇(θ − k)− ζ|

p

Kρ

∇θ · ∇ζ ζ p−1 (θ − k)− p

Kρ

(θ − k)p− |∇ζ| ,

(7.8)

using Young’s inequality. Since t ∈ (−τ, 0) is arbitrary, we can combine estimates (7.7) and (7.8) to obtain the next result. Proposition 7.2. Let θ be an approximate solution of (7.1) and let k < M . There exists a constant C, that is independent of , such that for every cylinder Q(τ, ρ) ⊂ ΩT , 0 (θ − k)2− ζ p dx + |∇(θ − k)− ζ|p dx dt sup −τ

≤C

−τ Kρ 0

Kρ ×{t}

Kρ ×{−τ }

(θ − k)− ζ p dx + C

0

Kρ

(θ − k)p− |∇ζ|p dx dt

(θ − k)− ζ p−1 ζt dx dt.

+C −τ

−τ

(7.9)

Kρ

The conclusion is that dealing with the singularity produced by the maximal monotone graph involves a regularization procedure and the search for estimates that are uniform with respect to that regularization, i.e., that are independent of the parameter . Since γ is not uniformly bounded above near the singularity, this leads to the appearance of integral terms in the energy estimates involving the L1 -norm of the solution, besides the L2 -norm. Thus, with three powers in play, the inhomogeneity in the estimates becomes more severe. The ultimate eﬀect will be the loss of the H¨older continuity in the limit, for which only continuity will be obtained via Ascoli’s theorem. When k > , we are above the singularity and the energy estimates for (θ − k)+ read as follows.

112

7 Phase Transitions: The Doubly Singular Stefan Problem

Proposition 7.3. Let θ be an approximate solution of (7.1) and k > . Then there exists a constant C, that is independent of , such that for every cylinder Q(τ, ρ) ⊂ ΩT ,

sup

−τ
Kρ ×{t}

≤C

Kρ ×{−τ }

(θ −

k)2+ ζ p

−τ Kρ 0

0

−τ

|∇(θ − k)+ ζ|p dx dt (θ − k)p+ |∇ζ|p dx dt

Kρ

(θ − k)2+ ζ p−1 ζt dx dt.

+C −τ

dx +

(θ − k)2+ ζ p dx + C

0

(7.10)

Kρ

Concerning the logarithmic estimates, we obtain the following results. The reasoning is that of Section 2.3. Proposition 7.4. Let θ be an approximate solution of (7.1), k ∈ R and 0 < c < Hθ− ,k . There exists a constant C > 0, that is independent of , such that for every cylinder Q(τ, ρ) ⊂ ΩT , − 2 ψ (θ ) ζ p dx sup −τ

≤

Kρ ×{−τ }

0

2γ (s)ψ − (s)(ψ − ) (s)

k

ζ p dx

ds +

ψ − (θ )|(ψ − ) (θ )|2−p |∇ζ|p dx dt.

+C −τ

Kρ ×{t}

θ

(7.11)

Kρ

Remark 7.5. In this estimate there is a dependence on through γ . We will see later how to overcome this diﬃculty. Proposition 7.6. Let θ be an approximate solution of (7.1), k > and 0 < c < Hθ+ ,k . There exists a constant C > 0, that is independent of , such that for every cylinder Q(τ, ρ) ⊂ ΩT , + + 2 p 2 ψ (θ ) ζ dx ≤ ψ (θ ) ζ p dx sup −τ
Kρ ×{t} 0

+C −τ

Kρ ×{−τ }

2−p ψ + (θ ) (ψ + ) (θ ) |∇ζ|p dx dt.

(7.12)

Kρ

7.3 The Intrinsic Geometry The study of the interior regularity of the approximate solution, namely showing that it is continuous independently of , requires the choice of the right intrinsic geometry that somehow reﬂects the two singularities in the equation.

7.3 The Intrinsic Geometry

113

To fully understand what is at stake, let us observe that a bridge between the singularity in time and a degeneracy in space can be made through rewriting equation (7.1) in terms of a v ∈ γ(u), taking into account that u = γ −1 (v) and γ −1 is now a well deﬁned function, such that γ −1 (s) = 0 for 0 ≤ s ≤ λ. It is clear that the time singularity in the u-equation becomes a space degeneracy for the v-equation. In the case of equation (7.1) with p > 2, which was treated in [51], we are in the presence of two types of degeneracy in the principal part of the equation and that explains why a rescaling in time is enough. Also, in the case 1 < p < 2 but with no jumps (i.e., for γ(s) = s), there is only a singularity, so a rescaling in space suﬃces (see [14, Ch. 4]). Here, we have the equivalent of both a singularity and a degeneracy in the principal part and so we need both rescalings, in space and in time. In order to simplify the notation, from now on we drop the subscript in θ . Given R > 0, suﬃciently small such that 1 Q R, R 2 ⊂ ΩT , deﬁne θ ; µ− := ess inf 1

ω := ess osc θ = µ+ − µ− 1

µ+ := ess sup θ ; 1 Q(R,R 2

Q(R,R 2 )

Q(R,R 2 )

)

and construct the cylinder Q (a0 Rp , c0 R) ,

ω (1−p)(2−p)

a0 =

A

;

c0 =

ω p−2 B

,

(7.13)

where A = 2s and B = 2s¯, for some s, s¯ > 1 to be chosen. Observe that for p = 2, a0 = c0 = 1, and these are the standard parabolic cylinders, reﬂecting the natural homogeneity of the space and time variables. We assume, without loss of generality, that ω ≤ 1 and also that 1 1 (7.14) ω > 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
2(n+2)p d∗ Rp

ω p 2(n+2)p 2 Rp

p

(θ − kn )2− ξnp +

χ[θ
|∇(θ − kn )− ξn | Qn

ω 2(n+2)p

χ[θ
1−p ∗ −1 ω c−p + (d ) χ[θ
2

Now observe that, since 1 < p < 2, and (θ − kn )− ≤

ω 2,

118

7 Phase Transitions: The Doubly Singular Stefan Problem (1−p)(2−p)

(θ − kn )2− = (θ − kn )− ≥

ω (1−p)(2−p)

p(2−p)

(θ − kn )− 2 Introducing the level

p(2−p)

(θ − kn )−

(θ − kn )p−

(θ − kn )p− = d∗ (θ − kn )−

p(2−p)

(θ − kn )p− .

kn + kn+1 < kn , kn+1 < k¯n := 2 we have

(θ − Kc0 Rn

≥d

kn )2− ξnp

≥d

∗

∗

p(2−p)

Kc0 Rn

kn − k¯n

p(2−p)

(θ − kn )−

(θ − kn )p− ξnp

Kc0 Rn

(θ − k¯n )p− ξnp

ω p(2−p) 1 p(2−p) =d (θ − k¯n )p− ξnp . 2 2n+3 Kc0 Rn ∗

Consequently,

sup p −d∗ Rn
Kc0 Rn ×{t}

(θ − k¯n )p− ξnp

p(2−p) 1 2 + ∗ 2(n+3)p(2−p) |∇(θ − k¯n )− ξn |p d ω Qn

ω p 2(n+2)p ω p(p−2) −p (n+3)p(2−p) ≤C 2 c0 ∗ p 2 d R 2 ω p(p−2)+(1−p) + (d∗ )−1 χ[θ
x , c0

z=

t d∗

and deﬁne the new functions ¯ z) = θ(x, t), θ(y,

ξ¯n (y, z) = ξn (x, t).

7.4 Analyzing the Singularity in Time

119

Denoting the new variables again by (x, t) and deﬁning An =

¯ t) < kn , An (t) = x ∈ KRn : θ(x,

0

p −Rn

|An (t)| dt,

the above inequality reads sup p −Rn
KRn

(θ¯ − k¯n )p− ξ¯np

p(2−p) 2 (n+3)p(2−p) ∇(θ¯ − k¯n )− ξ¯n p + 2 p ω Q(Rn ,Rn ) p(2−p) 2 C ω p 2(3−p)pn p(2+3(2−p)) ≤ 2 + 2 An . ω 2 Rp B Recalling once again that ω ≤ 1, and the properties of ξn , we conclude that (θ¯ − k¯n )− p p V (Q(Rp ,R )) n+1

n+1

p(2−p) 2 C ω p 2(3−p)pn p(2+3(2−p)) ≤ 2 + 2 An . ω 2 Rp B Now, on the one hand, we have ω p 1 (θ¯ − k¯n )p− ≥ (k¯n − kn+1 )p An+1 = A p(n+3) n+1 p 2 2 Q(Rn+1 ,Rn+1 ) and, on the other hand, using Theorem 2.11, (θ¯ − k¯n )p− p Q(Rn+1 ,Rn+1 )

p p p ≤ C [θ¯ < k¯n ] ∩ Q(Rn+1 , Rn+1 ) d+p (θ¯ − k¯n )− V p (Q(Rp

n+1 ,Rn+1 ))

p d+p

≤ CAn

(θ¯ − k¯n )− p p V (Q(Rp

n+1 ,Rn+1 ))

Combining these two inequalities with the previous one, we get p(2−p) 1+ p 2 C 2p(5+3(2−p)) + 2 2(4−p)pn An d+p . An+1 ≤ p ω R B Deﬁning Yn = and noting that

An |Q(Rnp , Rn )|

120

7 Phase Transitions: The Doubly Singular Stefan Problem p

|Q(Rnp , Rn )|1+ d+p ≤ 2d+2p Rp p |Q(Rn+1 , Rn+1 )| we arrive at Yn+1

C ≤ ω

2 B

p(2−p)

p 1+ d+p

+ 2 2(4−p)pn Yn

.

Using Lemma 2.9, we conclude that if Y0 ≤

C ω

2 B

−(d+p) p

p(2−p)

2−p(4−p)(

+2

d+p p

2

)

then Yn → 0 as n → ∞. Since

2 B

d+p p

p(2−p)

≤2

+2

d p

d

≤ 2p

2 B

(2−p)(d+p) +2

d+p p

d+p 1+2 p ,

if we take Y0 ≤ C −

d+p p

2−p(4−p)(

d+p p

2

d ) ω d+p p 2− p

−1 d+p d+p 1+2 p = Cω p =: ν0

(7.22)

we obtain Yn → 0 as n → ∞, which implies that An → 0 as n → ∞. But 0 ¯ t) < µ− + ω dt x ∈ KR : θ(x, 2 p , Y0 = −R |Q(Rp , R)| so (7.22) is our hypothesis (7.20). Since Rn R2 and kn µ− + An → 0 as n → ∞, means that p R R ¯ t) ≤ µ− + ω = 0 (x, t) ∈ Q : θ(x, , 2 2 4

ω 4,

having

that is, going back to the original variables and function, p R ω R − ∗ . , c0 θ(x, t) > µ + , a.e. (x, t) ∈ Q d 4 2 2 Consider now the time level −t∗ = t¯ − d∗ ( R2 )p . From the conclusion of Lemma 7.9, we have θ(x, −t∗ ) > µ− +

ω , 4

a.e. x ∈ Kc0 R . 2

7.4 Analyzing the Singularity in Time

121

We will use this time level as an initial condition to bring the information up to t = 0, and therefore to obtain an analogous inequality in a full cylinder of the type Q(τ, c0 ρ). A ﬁrst step in this direction is given by the following result. Lemma 7.10. Assume that (7.17) holds for some t¯ ∈ I(ω) and for all x ¯∈ KR(ω) . Given ν1 ∈ (0, 1), there exists s1 ∈ N, depending only on the data and A, such that, if B ≥ 2s1 , then ω ∀t ∈ (−t∗ , 0). x ∈ Kc0 R4 : θ(x, t) < µ− + s1 < ν1 Kc0 R4 , 2 Proof. Consider the cylinder Q(t∗ , c0 R2 ) and write the logarithmic estimate (7.11) over this cylinder, for the function (θ − k)− , with k = µ− +

ω 4

and

c=

ω 2n+2

,

where n is to be chosen. Deﬁning − := ess sup (θ − k)− ≤ k − θ ≤ Hθ,k Q(t∗ ,c0 R 2 )

ω 4

− − If Hθ,k ≤ ω8 , the result is trivial for the choice s1 = 3. Assuming Hθ,k > ω8 , recall from section 2.3 that the logarithmic function ψ − = ψ − (θ), introduced in (2.7), is well deﬁned and satisﬁes the estimates

ψ − ≤ ln(2n ) = n ln 2, − 2−p

|(ψ ) |

= = ≤

because

1 − Hθ,k +θ−k+c

− Hθ,k

− Hθ,k − Hθ,k +θ−k+c

= 2n ;

2n+2

p(2−p) 1 − Hθ,k +θ−k+c n+2 p(2−p) 2 = (d∗ )−1 , ω

(p−1)(2−p) +θ−k+c

c

ω 4 ω

2−p

ω (p−1)(2−p) 1 p(2−p) 2

≤

− − + θ − k + c ≤ Hθ,k < ω2 and 0 < p − 1 < 1. since c ≤ Hθ,k Take as a cutoﬀ function x → ξ(x) (independent of t), deﬁned in Kc0 R , 2 and satisfying ⎧ ⎪ 0 ≤ ξ ≤ 1 in Kc0 R2 ⎪ ⎪ ⎪ ⎨ ξ ≡ 1 in Kc0 R 4 ⎪ ⎪ ⎪ ⎪ ⎩ |∇ξ| ≤ c04R .

122

7 Phase Transitions: The Doubly Singular Stefan Problem

The logarithmic estimate takes the form 0 − 2 p (ψ ) ξ ≤ C sup −t∗
Kc

−t∗

R 0 2

≤ Cn ln 2 (d∗ )−1

2n+2 ω

Kc

ψ − |(ψ − ) |2−p |∇ξ|p R 0 2

p(2−p)

4p Kc0 R2 t∗ ,

cp0 Rp

since θ(x, −t∗ ) > k in the cube Kc0 R which implies that 2

−

ψ (x, −t∗ ) = 0,

for x ∈ Kc0 R . 2

Recalling that t∗ < a0 Rp , and taking B ≥ 2n+2 , we get sup (ψ − )2 ξ p ≤ CA(p−1)(2−p) n Kc0 R . −t∗
Kc

4

R 0 2

We estimate from below the left hand side of the above inequality integrating over the smaller set ω x ∈ Kc0 R : θ(x, t) < µ− + n+2 ⊂ Kc0 R . 4 2 2 In this set, ξ ≡ 1 and − − − ω4 + ω4 Hθ,k Hθ,k − ≥ ln ≥ (n − 1) ln 2 ψ = ln − − ω Hθ,k +θ−k+c Hθ,k − ω4 + 2n+1 − since Hθ,k −

ω 4

≤ 0. Then, for all t ∈ (−t∗ , 0),

x ∈ Kc0 R4 : θ(x, t) < µ− +

ω n (p−1)(2−p) R . ≤ CA K c0 4 2n+2 (n − 1)2

n (p−1)(2−p) we get CA(p−1)(2−p) (n−1) Choosing n > 1 + 2C 2 < ν1 and the ν1 A result is proved for the choice s1 = n + 2.

Remark 7.11. Note that s1 can only depend on ω through A; it will be shown that A can be determined independently of ω. Observe also that the dependency appearing in the original logarithmic estimate has been overcome. We are now in position to prove the main result of this section. Proposition 7.12. There exists s1 ∈ N, depending only on the data and A, such that, if (7.17) holds for some t¯ ∈ I(ω) and for all x ¯ ∈ KR(ω) , then ω R . (7.23) θ(x, t) > µ− + s1 +1 , a.e. (x, t) ∈ Q t∗ , c0 2 8

7.4 Analyzing the Singularity in Time

123

Proof. Consider the decreasing sequence of real numbers Rn =

R R + n+3 , 8 2

n = 0, 1, . . .

and construct the family of nested and shrinking cylinders Qn = Q(t∗ , c0 Rn ), where t∗ is given as before. Write the energy estimates (7.9) for the functions (θ − kn )− over Qn , with kn = µ− +

ω ω + s1 +1+n , 2s1 +1 2

and choosing piecewise smooth cutoﬀ functions ξn (x) deﬁned in Kc0 Rn and satisfying, for n = 0, 1, 2, . . . , ⎧ 0 ≤ ξn ≤ 1 in Kc0 Rn ⎪ ⎪ ⎪ ⎪ ⎨ ξn ≡ 1 in Kc0 Rn+1 ⎪ ⎪ ⎪ ⎪ n+4 ⎩ |∇ξn | ≤ 2c0 R . Since, for all n = 0, 1, 2, . . ., θ(x, −t∗ ) > µ− +

ω > kn , 4

for x ∈ Kc0 R ⊃ Kc0 Rn 2

we have

K c0 R n

(θ(·, −t∗ ) − kn )− ξnp = 0,

and consequently the energy estimates read (θ − kn )2− ξnp + sup −t∗
Kc0 Rn

≤C Qn

|∇(θ − kn )− ξn |

p

Qn

(θ − kn )p− |∇ξn |p ≤ C

ω p 2(n+4)p χ[θ
Reasoning as in the proof of Lemma 7.9, we estimate from below the left hand side in the following way: letting kn+1 + kn < kn kn+1 < k¯n := 2 then, since 1 < p < 2, (θ − kn )− ≤ (θ − kn )2− ≥

ω 2 s1

and t∗ ≤ a0 Rp , we have

ω (1−p)(2−p) ( R )p t ω p(2−p) ∗ 2 2(n+3)p(p−2) (θ − k¯n )p− 2s1 t∗ ( R2 )p 2s1

124

7 Phase Transitions: The Doubly Singular Stefan Problem

≥ 2−p

ω (1−p)(2−p) ω (p−1)(2−p) ω p(2−p) t∗ 2(n+3)p(p−2) R p (θ − k¯n )p− 2s1 A 2s1 (2) s1 (p−1)(2−p) 2 ω p(2−p) (n+3)p(p−2) t∗ ≥ 2−p 2 (θ − k¯n )p− A B ( R2 )p (n+3)p(p−2) ≥ c−p 0 2

t∗ (θ R p (2)

− k¯n )p−

if we choose B ≥ 2s1 and s1 such that s1 > log2 A +

p . (p − 1)(2 − p)

Therefore the above integral inequality takes the form sup

−t∗
R p

( ) (θ − k¯n )p− ξnp + cp0 2(n+3)p(2−p) 2 t∗ Kc0 Rn ≤C

∇(θ − k¯n )− ξn p

Qn

R p ω p 2(3−p)pn 3p(3−p) ( 2 ) 2 χ[θ
Introducing the change of variables y=

x , c0

z=

R 2

p

t t∗

and deﬁning the new functions ¯ z) = θ(x, t), θ(y,

ξ¯n (y) = ξn (x),

we write the inequality in the new variables (again denoted by (x, t)), obtaining p ¯p p (n+3)p(2−p) ¯ ¯ ∇(θ¯ − k¯n )− ξ¯n p sup (θ − kn )− ξn + c0 2 p −( R 2 )
p Q(( R 2 ) ,Rn )

KRn

≤C

ω p 2(3−p)pn 3p(3−p) 2 χ[θ
Deﬁning An =

¯ t) < kn An (t) = x ∈ KRn : θ(x,

0

p −( R 2 )

|An (t)| dt,

and recalling the deﬁnition of c0 and that ω ≤ 1, we arrive at p ¯ θ − k¯n )− V p (Q(( R )p ,R 2

n+1 ))

≤C

ω p 2(3−p)pn 23p(3−p) An . 2s1 ( R2 )p

7.4 Analyzing the Singularity in Time

Now, since

ω p 1 An+1 = (k¯n − kn+1 )p An+1 2s1 2(n+3)p p p θ − k¯n )− V p (Q(( R )p ,R (θ¯ − k¯n )p− ≤ CAnd+p ¯

≤

2

p Q(( R 2 ) ,Rn+1 )

n+1 ))

125

,

we get p 2(4−p)pn 3p(4−p) 1+ d+p 2 A . n ( R2 )p

An+1 ≤ C Deﬁne Yn =

An . p |Q(( R 2 ) ,Rn )|

Due to the fact that p

|Q(( R2 )p , Rn )|1+ d+p ≤ 2d+p |Q(( R2 )p , Rn+1 )|

R 2

p

we get the algebraic inequality p 1+ d+p

Yn+1 ≤ C2(4−p)pn Yn so, as before, if Y0 ≤ C −

d+p p

2−p(4−p)(

d+p 2 p )

=: ν1

then Yn → 0 as n → ∞. Using Lemma 7.10 for this value of ν1 we conclude that there exists s1 ∈ N such that ω ∀t ∈ (−t∗ , 0). x ∈ Kc0 R4 : θ < µ− + s1 < ν1 Kc0 R4 , 2 To conclude, note that 0 ¯ t) < µ− + ω dt x ∈ K R4 : θ(x, 2s1 p −( R 2 ) Y0 = R p R |Q(( 2 ) , 4 )| =

( R2 )p t∗

ω x ∈ Kc0 R4 : θ(x, t) < µ− + s1 dt 2 −t∗ ≤ ν1 R p ( 2 ) |Kc0 R | 0

4

so Yn , An → 0 as n → ∞. Since Rn

R 8

and

we obtain θ(x, t) > µ− +

ω 2s1 +1

,

kn µ− +

ω 2s1 +1

,

R . a.e. (x, t) ∈ Q t∗ , c0 8

126

7 Phase Transitions: The Doubly Singular Stefan Problem

Corollary 7.13. Assume that (7.17) holds for some t¯ ∈ I(ω) and for all x ¯ ∈ KR(ω) . There exists a constant σ0 ∈ (0, 1), depending only on the data and A, such that θ ≤ σ0 ω. (7.24) ess osc ∗ p R Q(d ( R 2 ) ,c0 8 )

Proof. The proof is trivial, recalling that p p R R < −d∗ −t∗ = t¯ − d∗ 2 2 from which follows Q t∗ , c0 R8 ⊃ Q d∗ ( R2 )p , c0 R8 . We obtain σ0 = 1 −

1 . 2s1 +1

7.5 The Eﬀect of the Singularity in the Principal Part If the ﬁrst alternative does not hold then the second alternative is in force. We will show that, in this case, we can achieve a conclusion similar to (7.24). Note that the constant ν0 has already been determined and is given by (7.22). Lemma 7.14. Fix t¯ ∈ I(ω) and x ¯ ∈ KR(ω) for which (7.18) holds. There exists a time level ν0 (7.25) t0 ∈ t¯ − d∗ Rp , t¯ − d∗ Rp 2 such that 1 − ν0 ω + |Kd∗ R | . (7.26) x + Kd∗ R ] : θ(x, t0 ) > µ − ≤ x ∈ [¯ 2 1 − ν20 Proof. Suppose not. Then, for all t ∈ [t¯ − d∗ Rp , t¯ − ν20 d∗ Rp ], ω x, t¯) + Q(d∗ Rp , d∗ R)] : θ(x, t) > µ+ − (x, t) ∈ [(¯ 2 ν ∗ p t¯− 20 d R ω x + Kd∗ R ] : θ(x, τ ) > µ+ − dτ ≥ x ∈ [¯ 2 t¯−d∗ Rp 1 − ν0 ν0 ∗ p d R = (1 − ν0 ) |Q(d∗ Rp , d∗ R)| |Kd∗ R | 1 − > 1 − ν20 2 which contradicts (7.18).

7.5 The Eﬀect of the Singularity in the Principal Part

127

This result tells us that, at the time level t0 , the portion of the cube [¯ x + Kd∗ R ] where θ(x) is near its supremum is small. In what follows we prove that the same happens at all time levels near the top of the cylinder [(¯ x, t¯) + Q(d∗ Rp , d∗ R)]. Lemma 7.15. There exists s2 ∈ N, depending only on the data and ω, such that ν 2 ω 0 + |Kd∗ R | , (7.27) x + Kd∗ R ] : θ(x, t) > µ − s2 < 1 − x ∈ [¯ 2 2 for all t ∈ (t0 , t¯). Proof. Assume, without loss of generality, that x ¯ = 0. Consider the logarithmic estimate (7.12) written over the cylinder Kd∗ R × (t0 , t¯), for the function (θ − k)+ , with ω ω and c = n∗ +1 , k = µ+ − 2 2 where n∗ is to be chosen. Assuming that the number n∗ has been chosen, we determine the length of the cube Kd∗ R by choosing ω p−2 d∗ = n∗ +1 . 2 In the deﬁnition of ψ + take + θ − k ≤ Hθ,k :=

ess sup Kd∗ R ×(t0 ,t¯)

(θ − k)+ ≤

ω . 2

+ + If Hθ,k ≤ ω4 , the result is trivial for the choice s2 = 2. Assuming Hθ,k > ω4 , + + and since Hθ,k − θ + k + c > 0, the logarithmic function ψ is well deﬁned and satisﬁes the estimates

ψ + ≤ n∗ ln 2,

ψ

2−p +

=

+ Hθ,k

since

+ Hθ,k + Hθ,k

−θ+k+c

(p−1)(2−p) −θ+k+c

≤ (d∗ )−1

2n∗ +1 ω

p(2−p)

≤

ω 2 ω

= 2n∗ ;

2n∗ +1

1 + Hθ,k −θ+k+c

p(2−p)

= (d∗ )−1 dp∗ ,

for the non trivial case θ > k + c. Take as a cutoﬀ function x → ξ(x), deﬁned in Kd∗ R , and satisfying ⎧ ⎪ ⎪ 0 ≤ ξ ≤ 1 in Kd∗ R ⎪ ⎪ ⎨ ξ ≡ 1 in K(1−σ)d∗ R , for some σ ∈ (0, 1) ⎪ ⎪ ⎪ ⎪ ⎩ −1 |∇ξ| ≤ (σd∗ R) .

128

7 Phase Transitions: The Doubly Singular Stefan Problem

The logarithmic estimates read + 2 p ψ sup ξ ≤ t0
t¯

and therefore sup

ψ+

ψ+

2

t0
ψ+

2

ξp

Kd∗ R ×{t0 }

+C t0

ψ+

2−p

|∇ξ|p

Kd∗ R

ξ p ≤ n2∗ (ln 2)2 |x ∈ Kd∗ R : θ(x, t0 ) > k + c| C + p p p n∗ ln 2 (d∗ )−1 dp∗ |Kd∗ R | (t¯ − t0 ) σ d∗ R

1 − ν0 n∗ + C |Kd∗ R | , ≤ n2∗ (ln 2)2 1 − ν20 σp

using Lemma 7.14, the estimates for ψ presented above, and the fact that t¯ − t0 ≤ d∗ Rp . In order to bound the left hand side from below, we integrate over the smaller set ω S = x ∈ K(1−σ)d∗ R : θ(x, t) > µ+ − n∗ +1 ⊂ K(1−σ)d∗ R 2 where ξ ≡ 1 and ψ + ≥ (n∗ − 1) ln 2, since + Hθ,k + Hθ,k − + because Hθ,k −

ω 2

ω 2

+

ω 2n ∗

=

+ − Hθ,k + Hθ,k −

ω ω 2 + 2 ω ω 2 + 2n ∗

≥ 2n∗ −1 ,

≤ 0. We obtain

|S| ≤

n∗ n∗ − 1

2

1 − ν0 1 − ν20

C + n∗ σ p

|Kd∗ R |.

Consequently, for all t ∈ (t0 , t¯), we have ω x ∈ Kd∗ R : θ(x, t) > µ+ − n∗ +1 ≤ |S| + Kd∗ R \ K(1−σ)d∗ R 2 ≤

≤ |S| + dσ|Kd∗ R | 2 n∗ 1 − ν0 C + + dσ |Kd∗ R |. n∗ − 1 1 − ν20 n∗ σ p

Choose σ so small that dσ ≤ 38 ν02 and then n∗ so large that 3 C ≤ ν02 n∗ σ p 8

and

n∗ n∗ − 1

2

ν0 (1 + ν0 ) =: β. ≤ 1− 2

7.5 The Eﬀect of the Singularity in the Principal Part

With these choices we obtain x ∈ Kd∗ R : θ(x, t) > µ+ −

ν 2 0 |Kd∗ R | , ≤ 1− 2n∗ +1 2 ω

129

∀t ∈ (t0 , t¯)

and the result follows with s2 = n∗ + 1.

3 2 Remark 7.16. From the choice σ ≤ 8d ν0 , we see that, in order to satisfy the ﬁrst condition, it suﬃces to choose the number n∗ such that −2(p+1)

n∗ ≥ Cν0

,

where C is a constant depending only on the data. From the second condition on n∗ we get √ β+ β , n∗ ≥ β−1 and, since β > 1 (and assuming, without loss of generality, that ν0 ∈ (0, 12 )), √ 2β 4 4 β+ β ≤ = +2≤ 2 +2 β−1 β−1 ν0 (1 − ν0 ) ν0 and it suﬃces to choose n∗ ≥

4 + 2. ν02

So n∗ is to be chosen such that

−2(p+1) 4 n∗ ≥ max Cν0 , 2 +2 . ν0

Recalling that ν0 = Cω

d+p p

, we choose

n∗ ≥ Cω −α ,

α=

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

and n∗ depends on the data and ω. Remark 7.17. This result determines the value s2 and consequently d∗ , which deﬁnes the size of the subcubes [¯ x + Kd∗ R ] making up the partition of the full cube Kc0 R . It thus have a double scope: to determine a level and a cylinder such that the measure of the set where θ is above such a level can be made small on that particular cylinder. Since for the diﬀerent values of t¯ we may get cylinders with diﬀerent axes, we now need to expand to a complete cylinder in space and then use the fact that t¯ is an arbitrary element of I(ω) to get a reduction of the oscillation in a smaller cylinder centered at the origin. In order to do so, and since the location ¯ of x ¯ within the cube KR(ω) is only known qualitatively, we will assume that x

130

7 Phase Transitions: The Doubly Singular Stefan Problem

is the centre of the larger cube [¯ x + K8c0 R ] which we assume to be contained in K 12 . Indeed, if that is not the case, there is nothing to prove since R

1

ω ≤ C(B, p) R 2(2−p) . We then work within the cylinder [¯ x + K8c0 R ] × (t0 , t¯) which is mapped into Q4 = K4 × (−4p , 0) through the change of variables t − t¯ x−x ¯ p , z=4 . y= t¯ − t0 2c0 R With this same mapping, the cube [¯ x + Kd∗ R ] is mapped into Kh0 , where s2 2−p h0 = 12 2B < 1. Deﬁne the new function n∗ 2 θ¯ = (θ − µ+ ) ω and observe that θ¯ ≤ 0 and that, for the new variables and function, (7.27) is now written in the form 2 ¯ z) < − 1 > ν0 |Kh | , ∀z ∈ (−4p , 0). y ∈ Kh : θ(y, (7.28) 0 0 2 2 7.5.1 An Equation in Dimensionless Form The new function satisﬁes, in the sense of the distributions, an equation similar to that satisﬁed by θ, namely (denoting again the new variables by (x, t)) ¯ − C div |∇θ| ¯ p−2 ∇θ¯ = 0 in D (Q4 ) (7.29) ∂t γ˜ (θ) ¯ = γ (θ) and where γ˜ is such that γ˜ (θ) 1 (p−1)(2−p) ω 23p Since t0 ∈ t¯ − d∗ Rp , t¯ − ν20 d∗ Rp , C=

1 C ∈ 3p 2

2n∗ +p−1 Bp

2−p

2n∗ Bp

2−p

ν0 1 , 2 23p

(t¯ − t0 ) . Rp

2n∗ +p−1 Bp

2−p ! .

In order to simplify the calculations we will assume, for the time being, that (7.30) ∂t γ˜ θ¯ ∈ C −4p , 0; L1 (K4 )

7.5 The Eﬀect of the Singularity in the Principal Part

131

and will remove this assumption later. The weak formulation of (7.29) is then given by p−2 ¯ ∇θ¯ ∂t γ˜ (θ)ϕ + C ∇θ¯ · ∇ϕ = 0, K4

K4

for all t ∈ (−4 , 0) and for all ϕ ∈ C(Q4 ) ∩ C(−4p , 0; W01,p (K4 )). Due to the fact that the equation is weakly parabolic and θ¯ is a solution, ¯ (θ − k)+ is a sub-solution and then, for all admissible test functions ϕ ≥ 0, ¯ ∇(θ¯ − k)+ p−2 ∇(θ¯ − k)+ · ∇ϕ ≤ 0, ∂t γ˜ (θ − k)+ ϕ + C p

K4

K4

for all t ∈ (−4p , 0). In this inequality we take ϕ=

ξp −k(1 − δ) − (θ¯ − k)+

p−1

where k and δ ∈ (−1, 0) are to be chosen and ξ(x, t) = ξ1 (x)ξ2 (t) is a piecewise smooth cutoﬀ function deﬁned in Q4 satisfying ⎧ ⎨ 0 ≤ ξ ≤ 1 in Q4 ; ξ ≡ 1 in Q2 ; ξ ≡ 0 on ∂p Q4 ; ⎩

|∇ξ1 | ≤ 1 ;

0 ≤ (ξ2 )t ≤ 1 ;

and the property that the sets {x ∈ K4 : ξ1 (x) > −k} are convex for all k ∈ (−1, 0). Set −k(1 − δ) ¯ ≥0 (7.31) ψk (θ) = ln −k(1 − δ) − (θ¯ − k)+ and

¯ = φk (θ) 0

¯ (θ−k) +

1 ds (−k(1 − δ) − s)p−1

(7.32)

and observe that 1. for k, δ ∈ (−1, 0), the functions ϕ, ψk and φk are well deﬁned since ⎧ ⎨ −k + kδ if θ¯ ≤ k ¯ −k(1 − δ) − (θ − k)+ = ≥0; ⎩ kδ − θ¯ if θ¯ > k 2. the graph γ has a jump at θ = 0 so the corresponding graph γ˜ has a jump at n∗ 2 + ¯ < 0. θ = −µ ω

132

7 Phase Transitions: The Doubly Singular Stefan Problem

2n ∗

< −1, i.e., ω n∗ > log2 µ+

If we choose n∗ such that −µ+

ω

then, for k ∈ (−1, 0), the function (θ¯− k)+ is above the singularity in time and γ˜ ≡ 1. Therefore, for

ω −α n∗ > max Cω , log2 µ+

and k, δ ∈ (−1, 0), the weak formulation presented above reads ∇(θ¯ − k)+ p−2 ∇(θ¯ − k)+ · ∇ϕ ≤ 0, ∂t (θ¯ − k)+ ϕ + C K4

(7.33)

K4

for all t ∈ (−4p , 0). The ﬁrst term of (7.33) can be estimated from below as follows ¯ p ∂t (θ¯ − k)+ ϕ = ∂t φk (θ)ξ K4 K4 d ¯ p−p ¯ p−1 ξt = φk (θ)ξ φk (θ)ξ dt K4 K4 d C 1 p ¯ − ≥ φk (θ)ξ dt K4 2−p using the conditions on ξ and the fact that 1

(−k(1 − δ))2−p − (−k(1 − δ) − (θ¯ − k)+ )2−p 2−p 22−p 1 (−k(1 − δ))2−p < , k, δ ∈ (−1, 0). < 2−p 2−p

¯ = φk (θ)

For the second term we have ∇(θ¯ − k)+ p−2 ∇(θ¯ − k)+ · ∇ϕ K4

=p K4

p−1 ξ ∇(θ¯ − k)+ p−2 ∇(θ¯ − k)+ · ∇ξ −k(1 − δ) − (θ¯ − k)+ p |∇(θ¯ − k)+ | ξp. +(p − 1) −k(1 − δ) − (θ¯ − k)+ K4

Recalling the deﬁnition of ψk , we have ¯ = ∇ψk (θ)

∇(θ¯ − k)+ −k(1 − δ) − (θ¯ − k)+

7.5 The Eﬀect of the Singularity in the Principal Part

133

and, using Young’s inequality (ρ is to be chosen), we get from the identity above, ∇(θ¯ − k)+ p−2 ∇(θ¯ − k)+ · ∇ϕ K4

≥ (p − 1) 1 − ρ

−p p−1

¯ p ξ p − ρp ∇ψk (θ)

K4

Taking ρ = 2 obtain

p−1 p

p

|∇ξ| . K4

, recalling the conditions on ξ and the bounds on C, we ﬁnally

d dt

˜ ¯ p ξ p ≤ C1 , ∇ψk (θ) 2−p K4

¯ p + C˜0 φk (θ)ξ K4

(7.34)

where C˜0 = C˜0 (p, d, n∗ , B, ω) and C˜1 = C˜1 (p, d, n∗ , B). Note that, for all t ∈ (−4p , 0), ¯ = 0] ∩ [ξ = 1] = [θ¯ ≤ k] ∩ K2 ≥ [θ¯ < k] ∩ Kh > 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

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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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