# American Institute of Mathematical Sciences

April  2014, 34(4): 1355-1374. doi: 10.3934/dcds.2014.34.1355

## Gradient flow structures for discrete porous medium equations

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

Received  December 2012 Revised  March 2013 Published  October 2013

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.
Citation: Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
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