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Gradient flow structures for discrete porous medium equations
1. | University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany, Germany |
References:
[1] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008).
|
[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. |
[4] |
H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland Mathematics Studies, (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., (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. |
[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. |
[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. |
[10] |
N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879.
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.
doi: 10.1137/S0036141096303359. |
[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. |
[13] |
R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153.
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.
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.
doi: 10.1088/0951-7715/24/4/016. |
[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. |
[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. |
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).
|
[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. |
[4] |
H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland Mathematics Studies, (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., (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. |
[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. |
[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. |
[10] |
N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics,, SIAM J. Math. Anal., 45 (2013), 879.
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.
doi: 10.1137/S0036141096303359. |
[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. |
[13] |
R. J. McCann, A convexity principle for interacting gases,, Adv. Math., 128 (1997), 153.
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.
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.
doi: 10.1088/0951-7715/24/4/016. |
[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. |
[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. |
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