Gradient flow structures for discrete porous medium equations

Matthias Erbar; Jan Maas

doi:10.3934/dcds.2014.34.1355

Article Contents

2014, Volume 34, Issue 4: 1355-1374. Doi: 10.3934/dcds.2014.34.1355

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Gradient flow structures for discrete porous medium equations

Matthias Erbar^1, and
Jan Maas^1,

1.
University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn

Received: December 2012

Revised: March 2013

Published: April 2014

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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.

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Mathematics Subject Classification: Primary: 49Q20; Secondary: 76S05.

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