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The Porous Medium Equation THE POROUS MEDIUM EQUATION Mathematical theory by JUAN LUIS VAZQUEZ¶ Dpto. de Matem¶aticas Univ. Auton¶omade Madrid 28049 Madrid, SPAIN ||||||||- i To my wife Mariluz Preface The Heat Equation is one of the three classical linear partial di®erential equations of second order that form the basis of any elementary introduction to the area of Partial Di®erential Equations. Its success in describing the process of thermal propagation has known a per- manent popularity since Fourier's essay Th¶eorieAnalytique de la Chaleur was published in 1822, [237], and has motivated the continuous growth of mathematics in the form of Fourier analysis, spectral theory, set theory, operator theory, and so on. Later on, it contributed to the development of measure theory and probability, among other topics. The high regard of the Heat Equation has not been isolated. A number of related equa- tions have been proposed both by applied scientists and pure mathematicians as objects of study. In a ¯rst extension of the ¯eld, the theory of linear parabolic equations was devel- oped, with constant and then variable coe±cients. The linear theory enjoyed much progress, but it was soon observed that most of the equations modelling physical phenomena without excessive simpli¯cation are nonlinear. However, the mathematical di±culties of building theories for nonlinear versions of the three classical partial di®erential equations (Laplace's equation, heat equation and wave equation) made it impossible to make signi¯cant progress until the 20th century was well advanced. And this observation applies to other important nonlinear PDEs or systems of PDEs, like the Navier-Stokes equations. The great development of Functional Analysis in the decades from the 1930's to the 1960's made it possible for the ¯rst time to start building theories for these nonlinear PDEs with full mathematical rigor. This happened in particular in the area of parabolic equations where the theory of Linear and Quasilinear Parabolic Equations in divergence form reached a degree of maturity reflected for instance in the classical books of Ladyzhenskaya et al. [357] and Friedman [239]. The aim of the present text is to provide a systematic presentation of the mathematical theory of the nonlinear heat equation m (PME) @tu = ¢(u ); m > 1 ; usually called the Porous Medium Equation (shortly, PME), posed in the d-dimensional Euclidean space, with interest in the cases d = 1; 2; 3 for the applied scientist, with no dimension restriction for the mathematician. ¢ = ¢x represents the Laplace operator m¡1 acting on the space variables. We will also study the complete form, ut = ¢(juj u) + f; but in a less systematic way. Other variants appear in the literature but will be given less attention, since we keep to the idea of presenting a rather complete account of the main results and methods for the basic PME. The reader may wonder why such a simple-looking variation of the famous and well- i ii PME known heat equation (HE): ut = ¢u, needs a book of its own. There are several answers to this question: the theory and properties of the PME depart strongly from the heat equa- tion; it contains interesting and sometimes sophisticated developments of nonlinear analysis; there are a number of interesting applications where this theory, with all its di®erences, is necessary and useful; and, ¯nally, similar treatises have been written for individual equa- tions with a strong personality. As for the latter argument, we have the example of the heat equation itself, described in the monographs by Cannon [148] and Widder [525], and also the Stefan Problem, that is closely related to the HE and the PME, and was reported about in the books of Cannon [148], Rubinstein [454] and Meirmanov [388]. Let us now comment on the ¯rst aspects listed some lines above. The theory that has been developed and we present in this text not only settles the main problems of existence, uniqueness, stability, smoothness, dynamical properties and asymptotic behavior. In doing so, it contributes a wealth of new ideas with respect to the heat equation; great novelties occur also with respect to the standard nonlinear theories, represented by the theory of nonlinear parabolic equations in divergence form to which the Porous Medium Equation belongs. This is due to the fact that the equation is not parabolic at all points, but only degenerate parabolic, a fact that has deep mathematical consequences, both qualitative and quantitative. On the other hand, and as a sort of compensation, the equation enjoys a number of nice properties due to its simple form, like scaling invariance. This aspect makes the PME an interesting benchmark in the development of nonlinear analytical tools for the quite general classes of nonlinear, formally parabolic equations that continue to make their way into the pure and applied sciences, and then into the mainstream of mathematics. There are a number of physical applications where the simple PME model appears in a natural way, mainly to describe processes involving fluid flow, heat transfer or di®usion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other ¯elds. All of these reasons support the interest of its study both for the mathematician and the scientist. Context In spite of the simplicity of the equation and of having some important applications, and due perhaps to its nonlinear and degenerate character, the mathematical theory of the PME has been only gradually developed in the last decades after the seminal paper of Oleinik et al. [408] in 1958; in the 1980's the theory was ¯nally on ¯rm ground and has been rounded up since then. The idea of the book arose out of the participation of the author in this progress in the last three decades. The immediate motivation for writing the text is the feeling that the time is ripe for a reasonably complete version of the mathematics of the PME, once the main mathematical issues have come to be fairly well understood, and every result receives a proof in the style of Analysis. We are also aware of the need for researchers to apply to more complex models the wealth of techniques that work so well here, hence the need for clear and balanced expositions to learn the material. Therefore, we aim at providing a description of the questions of existence, uniqueness and the main properties of the solutions, whereby everything is derived from basic estimates using standard Functional Analysis and well- known PDE results. And we have tried to provide sound physical foundations throughout. iii Acknowledgments I am happy to acknowledge my deep indebtedness to D. Aronson, G. Barenblatt and L. Ca®arelli for sharing with me their expertise in the ¯eld and for so many other reasons, and to H. Brezis who initiated me many years ago in the ¯eld of nonlinear PDEs and encouraged me to write this book. I am also especially grateful to the late Ph. B¶enilan and to V. Galaktionov, S. Kamin and L. Peletier, since I learned so many things from them about this subject. I thank L. C. Evans for valuable advice. This work would not have been possible without the scienti¯c contributions and per- sonal help of my former students Ana Rodr¶³guez,Arturo de Pablo, Cecilia Yarur, Fernando Quir¶os,Guillermo Reyes, Juan Ram¶onEsteban, Manuela Chaves, Omar Gil, and Ra¶ulFer- reira to whom I would like to add Emmanuel Chasseigne, Matteo Bonforte, and Cristina BrÄandle,who was in charge of the graphical section; we did the computations with MAT- LAB. M. Portilheiro and A. S¶anchez also read parts of the book and made valuable sug- gestions. Numerous colleagues in the Department of Mathematics of UAM and elsewhere have helped in the work that led to this project at di®erent stages. I will take the liberty of mentioning only some of them: L. Boccardo, J. A. Carrillo, M. Crandall, M. A. Herrero, J. Hulshof, J. King, K. A. Lee, I. Peral, L. V¶eron,M. Walias. To them and many others, my deepest thanks. Finally, this work would have been completely impossible without the constant help and encouragement of my wife Mariluz and my two children, Isabel and Miguel. iv PME Contents Preface i 1 Introduction 1 1.1 The subject . 1 1.1.1 The porous medium equation . 1 1.1.2 The PME as a nonlinear parabolic equation . 2 1.2 Peculiar features of the PME . 4 1.2.1 Finite propagation and free boundaries . 4 1.2.2 The role of special solutions . 5 1.3 Nonlinear Di®usion. Related equations . 6 1.4 Contents . 8 1.4.1 The main problems and the classes of solutions . 8 1.4.2 Chapter overview . 9 1.4.3 What is not covered . 11 1.5 Reading the book . 12 Notes . 13 PART ONE 14 2 Main applications 15 2.1 Gas flow through a porous medium . 15 2.1.1 Extensions . 17 2.2 Nonlinear heat transfer . 17 2.3 Groundwater flow. Boussinesq's equation . 19 2.4 Population dynamics . 20 2.5 Other applications and equations . 21 2.6 Images, concepts and names taken from the applications . 21 Notes . 22 3 Preliminaries and Basic Estimates 25 v vi PME 3.1 Quasilinear equations and the PME . 25 3.1.1 Existence of classical solutions . 26 3.1.2 Weak theories and the PME . 26 3.2 The GPME with good ©. Main estimates . 28 3.2.1 Maximum Principle and Comparison . 29 3.2.2 Other boundedness estimates . 30 3.2.3 The stability estimate. L1-contraction . 31 3.2.4 The energy identity .
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