Gradient flow structures for discrete porous medium equations

Matthias Erbar,Jan Maas

doi:10.3934/dcds.2014.34.1355

Abstract

We consider discrete porous medium equations of the form $\partial_t\rho_t = \Delta \phi(\rho_t)$, where $\Delta$ is the generator of a reversible continuous time Markov chain on a finite set $\boldsymbol{\chi} $, and $\phi$ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in $\mathbb{R}^n$ discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.

Highlights

It has been shown that discretisations of heat equations and Fokker-Planck equations can be formulated as gradient flows of the entropy with respect to a non-local transportation metric W on the space of probability measures
The above-mentioned results can be regarded as discrete counterparts to the classical result of Jordan, Kinderlehrer and Otto [10], who showed that Fokker-Planck equations on Rn are gradient flows of the entropy with respect to the L2-Wasserstein metric on the space of probability measures
One of the most prominent examples is the porous medium equation, which has been identified as the Wasserstein gradient flow of the Renyi entropy in Otto’s seminal paper [16]

Summary

Introduction

It has been shown that discretisations of heat equations and Fokker-Planck equations can be formulated as gradient flows of the entropy with respect to a non-local transportation metric W on the space of probability measures. We are interested in more general gradient flow structures in the spirit of the Wasserstein gradient flow structure for the porous medium equation [16] For this purpose, for suitable (see Assumption 3.3) strictly convex functions f : [0, ∞) → R, we consider the entropy functional F : P(X ) → R defined by. Since the discrete transportation metrics discussed in this paper take over the role of the L2-Wasserstein metric in a discrete setting, it seems natural to ask whether they converge to the L2Wasserstein metric by a suitable limiting procedure First results in this spirit have been obtained in [9], where it was proved that discrete transportation metrics WN associated with simple random walk on the discrete torus (Z/N Z)d converge to the L2-Wasserstein metric W2 over the torus Td. The weight function considered in [9] is the logarithmic mean. Since this result can be obtained by a minor modification of the proof in [9], we do not give a detailed proof here, but merely point out the crucial properties which make the argument go through

Preliminaries on non-local transportation metrics

Discrete porous medium equations as gradient flows of the entropy

Geodesic κ-convexity of entropy functionals

Gromov-Hausdorff convergence