Gradient flow structures for discrete porous medium equations

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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

Abstract
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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 and Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.
[2]	L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, arXiv:1109.0222, (2012).
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[4]	H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[5]	E. Carlen and J. Maas, An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy, to appear in Comm. Math. Phys., arXiv:1203.5377, (2012).
[6]	S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6.
[7]	S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X.
[8]	M. Erbar, Gradient flows of the entropy for jump processes, to appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1204.2190, (2012).
[9]	M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.
[10]	N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315.
[11]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[12]	J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.
[13]	R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.
[14]	A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.
[15]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.
[16]	A. Mielke, Dissipative quantum mechanics using GENERIC, To appear in Proc. of the conference "Recent Trends in Dynamical Systems,'' (2013).
[17]	F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[18]	F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

show all references

References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.
[2]	L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, arXiv:1109.0222, (2012).
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[4]	H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[5]	E. Carlen and J. Maas, An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy, to appear in Comm. Math. Phys., arXiv:1203.5377, (2012).
[6]	S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6.
[7]	S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X.
[8]	M. Erbar, Gradient flows of the entropy for jump processes, to appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1204.2190, (2012).
[9]	M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.
[10]	N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315.
[11]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[12]	J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.
[13]	R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.
[14]	A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.
[15]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.
[16]	A. Mielke, Dissipative quantum mechanics using GENERIC, To appear in Proc. of the conference "Recent Trends in Dynamical Systems,'' (2013).
[17]	F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[18]	F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.

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