The porous medium equation: mathematical theory

647 Pages·2007·3.066 MB·English

by  Juan Luis Vazquez

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V´azquez: Theporousmediumequation
The Porous
Medium Equation
Mathematical Theory
´
JUAN LUIS VAZQUEZ
Dpto. de Matem´aticas
Univ. Auton´oma de Madrid
28049 Madrid, SPAIN
CLARENDON PRESS • OXFORD
2007
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PREFACE
Theheatequationisoneofthethreeclassicallinearpartialdifferentialequations
of second order that form the basis of any elementary introduction to the area
of partial differential equations. Its success in describing the process of thermal
propagation has known a permanent popularity since Fourier’s essay Th´eorie
Analytique 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 equations have been proposed both by applied scientists and pure
mathematicians as objects of study. In a first extension of the field, the theory
of linear parabolic equations was developed, with constant and then variable
coefficients. The linear theory enjoyed much progress, but it was soon observed
that most of the equations modelling physical phenomena without excessive
simplification are nonlinear. However, the mathematical difficulties of building
theoriesfornonlinearversionsofthethreeclassicalpartialdifferentialequations
(Laplace’sequation,theheatequationandthewaveequation)madeitimpossible
to make significant progress until the twentieth century was well advanced. And
thisobservationappliestootherimportantnonlinearPDEsorsystemsofPDEs,
like the Navier–Stokes equations.
The great development of functional analysis in the decades from the 1930s
tothe1960smadeitpossibleforthefirsttimetostartbuildingtheoriesforthese
nonlinearPDEswithfullmathematicalrigour.Thishappenedinparticularinthe
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
∂ u=∆(um), m>1, (PME)
t
usually called the porous medium equation (PME), posed in d-dimensional
Euclidean space, with interest in the cases d=1,2,3 for the applied scientist,
with no dimension restriction for the mathematician. ∆=∆ represents the
x
Laplace operator acting on the space variables. We will also study the complete
form, u =∆(|u|m−1u)+f, but in a less systematic way. Other variants appear
t
in the literature but will be given less attention, since we keep to the idea of
vii
viii
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
andwell-knownheatequation(HE):u =∆u,needsabookofitsown.Thereare
t
several answers to this question: the theory and properties of the PME depart
strongly from the heat equation; it contains interesting and sometimes sophis-
ticated developments of nonlinear analysis; there are a number of interesting
applications where this theory, with all its differences, is necessary and useful;
and, finally, similar treatises have been written for individual equations with a
strong personality. As for the latter argument, we have the example of the heat
equation itself,describedin themonographsbyCannon [148]and Widder[525],
and also the Stefan problem that is closely related to the HE and the PME and
was reported in the books of Cannon [148], Rubinstein [454] and Meirmanov
[388].
Let us now comment on the first 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 behaviour. 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,afactthathasdeepmathematicalconsequences,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,formallyparabolicequationsthatcontinuetomaketheirwayintothe
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 diffusion. Other applications have been proposed in mathematical
biology,lubrication,boundarylayertheory,andotherfields.Allofthesereasons
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 1980s the
theory was finally on firm 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
Preface ix
feelingthatthetimeisripeforareasonablycompleteversionofthemathematics
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
thesolutions,wherebyeverythingisderivedfrombasicestimatesusingstandard
functional analysis and well-known PDE results. And we have tried to provide
sound physical foundations throughout.
Acknowledgments
IamhappytoacknowledgemydeepindebtednesstoD.Aronson,G.Barenblatt
and L.Caffarelli for sharing with me their expertise in the field and for somany
other reasons, and to H. Brezis who introduced me many years ago to the field
of nonlinear PDEs and encouraged me to write this book. I am also especially
gratefultothelatePh.B´enilanandtoV.Galaktionov,S.KaminandL.Peletier,
sinceIlearnedsomanythingsfromthemaboutthissubject.IthankL.C.Evans
for valuable advice.
This work would not have been possible without the scientific contributions
andpersonalhelpofmyformerstudentsAnaRodr´ıguez,ArturodePablo,Cecilia
Yarur, Fernando Quiro´s, Guillermo Reyes, Juan Ram´on Esteban, Manuela
Chaves, Omar Gil, and Rau´l Ferreira to whom I would like to add Emmanuel
Chasseigne, Matteo Bonforte, and Cristina Bra¨ndle, who was in charge of the
graphical section; we did the computations with MATLAB. M. Portilheiro and
A.Sa´nchezalsoreadpartsofthebookandmadevaluablesuggestions.Numerous
colleaguesintheDepartmentofMathematicsofUAMandelsewherehavehelped
in the work that led to this project at different 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 con-
stant help and encouragement of my wife Mariluz and my two children Isabel
and Miguel.