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

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, 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.
Keywords: Rényi entropy, Non-local transportation metrics, porous medium equations..
Mathematics Subject Classification: Primary: 49Q20; Secondary: 76S0.
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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show all references

References:
[1]

Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

arXiv:1109.0222, (2012). Google Scholar

[3]

Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.  Google Scholar

[4]

North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.  Google Scholar

[5]

to appear in Comm. Math. Phys., arXiv:1203.5377, (2012). Google Scholar

[6]

Arch. Ration. Mech. Anal., 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6.  Google Scholar

[7]

SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X.  Google Scholar

[8]

to appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1204.2190, (2012). Google Scholar

[9]

Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.  Google Scholar

[10]

SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315.  Google Scholar

[11]

SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[12]

J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.  Google Scholar

[13]

Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.  Google Scholar

[15]

Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[16]

To appear in Proc. of the conference "Recent Trends in Dynamical Systems,'' (2013). Google Scholar

[17]

Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[18]

J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.  Google Scholar

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