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Gradient flow structures for discrete porous medium equations. (English) Zbl 1275.49084

Summary: 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 \(\mathbf\chi\), 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
76S05 Flows in porous media; filtration; seepage
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