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 - A, 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, (2008).   Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below,, , (2012).   Google Scholar

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375.  doi: 10.1007/s002110050002.  Google Scholar

[4]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland Mathematics Studies, (1973).   Google Scholar

[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., (2012).   Google Scholar

[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.  doi: 10.1007/s00205-011-0471-6.  Google Scholar

[7]

S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance,, SIAM J. Math. Anal., 40 (2008), 1104.  doi: 10.1137/08071346X.  Google Scholar

[8]

M. Erbar, Gradient flows of the entropy for jump processes,, to appear in Ann. Inst. Henri Poincaré Probab. Stat., (2012).   Google Scholar

[9]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy,, Arch. Ration. Mech. Anal., 206 (2012), 997.  doi: 10.1007/s00205-012-0554-z.  Google Scholar

[10]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879.  doi: 10.1137/120886315.  Google Scholar

[11]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[12]

J. Maas, Gradient flows of the entropy for finite Markov chains,, J. Funct. Anal., 261 (2011), 2250.  doi: 10.1016/j.jfa.2011.06.009.  Google Scholar

[13]

R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153.  doi: 10.1006/aima.1997.1634.  Google Scholar

[14]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains,, Calc. Var. Partial Differential Equations, 48 (2013), 1.  doi: 10.1007/s00526-012-0538-8.  Google Scholar

[15]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[16]

A. Mielke, Dissipative quantum mechanics using GENERIC,, To appear in Proc. of the conference, (2013).   Google Scholar

[17]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101.  doi: 10.1081/PDE-100002243.  Google Scholar

[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.  doi: 10.1006/jfan.1999.3557.  Google Scholar

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, (2008).   Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below,, , (2012).   Google Scholar

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375.  doi: 10.1007/s002110050002.  Google Scholar

[4]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland Mathematics Studies, (1973).   Google Scholar

[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., (2012).   Google Scholar

[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.  doi: 10.1007/s00205-011-0471-6.  Google Scholar

[7]

S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance,, SIAM J. Math. Anal., 40 (2008), 1104.  doi: 10.1137/08071346X.  Google Scholar

[8]

M. Erbar, Gradient flows of the entropy for jump processes,, to appear in Ann. Inst. Henri Poincaré Probab. Stat., (2012).   Google Scholar

[9]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy,, Arch. Ration. Mech. Anal., 206 (2012), 997.  doi: 10.1007/s00205-012-0554-z.  Google Scholar

[10]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879.  doi: 10.1137/120886315.  Google Scholar

[11]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[12]

J. Maas, Gradient flows of the entropy for finite Markov chains,, J. Funct. Anal., 261 (2011), 2250.  doi: 10.1016/j.jfa.2011.06.009.  Google Scholar

[13]

R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153.  doi: 10.1006/aima.1997.1634.  Google Scholar

[14]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains,, Calc. Var. Partial Differential Equations, 48 (2013), 1.  doi: 10.1007/s00526-012-0538-8.  Google Scholar

[15]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[16]

A. Mielke, Dissipative quantum mechanics using GENERIC,, To appear in Proc. of the conference, (2013).   Google Scholar

[17]

F. Otto, The geometry of dissipative evolution equations: The porous medium equation,, Comm. Partial Differential Equations, 26 (2001), 101.  doi: 10.1081/PDE-100002243.  Google Scholar

[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.  doi: 10.1006/jfan.1999.3557.  Google Scholar

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[3]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[4]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[5]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[6]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[7]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[8]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[9]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[10]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[11]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[12]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[13]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[15]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[16]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[17]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[18]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[19]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[20]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]