SEMIDISCRETIZATION AND LONG-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS

We review several results concerning the long-time asymptotics of nonlinear difiusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear difiu- sion models are proposed and their numerical properties analyzed. We demonstrate the long-time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear difiusion equations the long-time asymptotics can be characterized in terms of flxed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of flxed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities near zero.

[1]  Marco Di Francesco,et al.  Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for ut = Δϕ(u) , 2006 .

[2]  J. A. Carrillo,et al.  Asymptotic L1-decay of solutions of the porous medium equation to self-similarity , 2000 .

[3]  G. Toscani.  Kinetic and Hydrodynamic Models of Nearly Elastic Granular Flows , 2004 .

[4]  Barry F. Knerr,et al.  The porous medium equation in one dimension , 1977 .

[5]  A. Arnold,et al.  On generalized Csiszár-Kullback inequalities , 2000 .

[6]  Giuseppe Toscani,et al.  Strict contractivity of the 2-wasserstein distance for the porous medium equation by mass-centering , 2006 .

[7]  Ansgar Jüngel,et al.  Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities , 2001 .

[8]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[9]  Juan Luis Vázquez,et al.  Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium , 1983 .

[10]  M. Agueh.  Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. , 2002, math/0309410.

[11]  Juan Luis Vázquez,et al.  Asymptotic behaviour for the porous medium equation posed in the whole space , 2003 .

[12]  Giuseppe Toscani,et al.  WASSERSTEIN METRIC AND LARGE-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS , 2005 .

[13]  Anton Arnold,et al.  Entropy decay of discretized fokker-planck equations I—Temporal semidiscretization , 2003 .

[14]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[15]  Laurent Gosse,et al.  Identification of Asymptotic Decay to Self-Similarity for One-Dimensional Filtration Equations , 2006, SIAM J. Numer. Anal..

[16]  A. S. Kalashnikov.  Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations , 1987 .

[17]  J. Vázquez.  An Introduction to the Mathematical Theory of the Porous Medium Equation , 1992 .

[18]  Manuel del Pino,et al.  Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ , 2002 .

[19]  Ansgar Jüngel.  Numerical Approximation of a Drift‐Diffusion Model for Semiconductors with Nonlinear Diffusion , 1995 .

[20]  José A. Carrillo,et al.  Asymptotic Complexity in Filtration Equations , 2007 .

[21]  L. Véron.  Effets régularisants de semi-groupes non linéaires dans des espaces de Banach , 1979 .

[22]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .

[23]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[24]  M. Crandall,et al.  Uniqueness of Solutions of the Initial-Value Problem for u sub t - delta phi(u) = 0. , 1978 .

[25]  Piotr Biler,et al.  Intermediate asymptotics in L1 for general nonlinear diffusion equations , 2002, Appl. Math. Lett..

[26]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[27]  E. Caglioti,et al.  A kinetic equation for granular media , 2009 .

[28]  J. Vázquez.  The Porous Medium Equation. New Contractivity Results , 2005 .

[29]  W. Gangbo,et al.  Constrained steepest descent in the 2-Wasserstein metric , 2003, math/0312063.

[30]  M. Pierre.  Uniqueness of the solutions of ut−Δϕ(u) = 0 with initial datum a measure☆ , 1982 .

[31]  J. Carrillo,et al.  Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations , 2006 .

[32]  Giuseppe Toscani,et al.  A central limit theorem for solutions of the porous medium equation , 2005 .

[33]  C. Villani.  Topics in Optimal Transportation , 2003 .

[34]  C. Villani,et al.  ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS , 2004 .

[35]  Giuseppe Toscani,et al.  Finite speed of propagation in porous media by mass transportation methods , 2004 .