We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

/pdf/the-geometry-of-dissipative-evolution-equations-the-porous-4be4l1zzoc.pdf

The geometry of dissipative evolution equations: the porous medium equation

Introduction Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity Holder regularity The Dirichlet problem for concave equations Bibliography Index.

/pdf/fully-nonlinear-elliptic-equations-3l9epg4e5f.pdf

Fully Nonlinear Elliptic Equations

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

/pdf/nonlinear-partial-differential-equations-with-applications-204hhsvanv.pdf

Nonlinear partial differential equations with applications

CONTENTSIntroduction Chapter I. Existence and uniqueness of generalized solutions of the Cauchy problem and of initial boundary-value problems ??1.1. Theorems of existence and non-existence ??1.2. Uniqueness theorems Chapter II. Estimates and differential properties of generalized solutions ??2.1. The character of the dependence on the data ??2.2. Regularity and non-regularity Chapter III. Behaviour of generalized solutions for small values of time ??3.1. Finite rate of propagation of perturbations ??3.2. Inertia ??3.3. Shrinking of the size of the support Chapter IV. Behaviour of generalized solutions for large values of time ??4.1. Stabilization in finite time ??4.2. Localization of perturbations ??4.3. Asymptotic formulae References

http://iopscience.iop.org/article/10.1070/RM1987v042n02ABEH001309/pdf

Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

The Obstacle Problem Revisited.

The structure conditions are the ellipticity of a and the (weak) monotonicity of b, and b has to be a subgradient in case m > 1. First we treat the case that b is continuous, and later (Sect. 4) we include Stefan problems, that is, we allow b to have jumps. The special cases of an elliptic equation with time as parameter, that is, b(z)= 0, and the standard parabolic equation, that is, b(z)=z are included. Some special single equations of mixed elliptic and parabolic type are given in the following. The gas flow through a porous medium is described by the equation

Quasilinear elliptic-parabolic differential equations

The problem of minimizing /(Vu|2 + q2(x)\2(v)) dx in an appropriate class of functions v is considered. Here q(x) ¥= 0 and A2(t>) = X2 if v 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\)-\22) across the free boundary (u ¥= 0). Regularity and various properties are established for the minimizer u and for the free boundary. Introduction. In this paper we consider the problem of minimizing J{v) = f(\ Vv\2 + q2(x)X2(v))dx, vGK, where q2(x) ¥= 0, (a22 ifu>0, and X2(u) is lower semicontinuous at v = 0; it is assumed that A2 > 0 and A = A2 — A22 =£ 0. The class K consists of all functions v in L'loc(S2), with Vu G L2(I2) such that v — u° on a given open subset S of 3I2, and s is a domain in R". The analogous problem for functions in K+ = {v Œ K, v > 0 a.e.} was studied in (1); in that paper it was proved that any (local) minimizer of J(v) in K+ is Lipschitz continuous and, if n — 2, the free boundary 8{w > 0} is analytic if q(x) is analytic. The present variational problem is motivated by applications to the now of two liquids in models of jets and cavities; these applications will be studied in other forthcoming papers (5,6). The present work is aimed at extending results of (1). In particular, we shall establish nondegeneracy theorems, the Lipschitz continuity of the solution, and some properties of the free boundary; for n = 2 the free boundary is proved to be continuously differentiable. A new and rather powerful tool introduced in this paper is the monotonicity formula (Lemma 5.1) asserting that, for a minimizer u, if u(x0) = 0 then

/pdf/variational-problems-with-two-phases-and-their-free-ffu98x6ghz.pdf

Variational problems with two phases and their free boundaries

Saturated-unsaturated flow of an incompressible fluid through a porous medium is considered in the case of time-dependent water levels. This corresponds to coupling the mass conservation law with a continuous constitutive condition between water content and pressure. An existence result for the corresponding weak formulation is proved. Finally we study the limit as the constitutive relation degenerates into a maximal monotone graph.

/pdf/on-nonstationary-flow-through-porous-media-568jn3e6i2.pdf

On nonstationary flow through porous media

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/a-free-boundary-problem-for-quasi-linear-elliptic-equations-32mhc22zit.pdf

Hans Wilhelm Alt

Papers

Quasilinear elliptic-parabolic differential equations

Variational problems with two phases and their free boundaries

On nonstationary flow through porous media

A free boundary problem for quasi-linear elliptic equations

Compressible flows of jets and cavities