We study existence and approximation of non-negative solutions of partial differential equations of the type
Classification : 35K55, 35K15, 35B40
Mots clés : nonlinear diffusion equations, parabolic equations, variable coefficient parabolic equations, gradient flows, Wasserstein distance, asymptotic behaviour
@article{COCV_2009__15_3_712_0, author = {Lisini, Stefano}, title = {Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {712--740}, publisher = {EDP-Sciences}, volume = {15}, number = {3}, year = {2009}, doi = {10.1051/cocv:2008044}, zbl = {1178.35201}, mrnumber = {2542579}, language = {en}, url = {www.numdam.org/item/COCV_2009__15_3_712_0/} }
Lisini, Stefano. Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 712-740. doi : 10.1051/cocv:2008044. http://www.numdam.org/item/COCV_2009__15_3_712_0/
[1] Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Adv. Differential Equations 10 (2005) 309-360. | MR 2123134 | Zbl 1103.35051
,[2] Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR 1387558 | Zbl 0957.49029
,[3] Transport equation and cauchy problem for non-smooth vector fields. Lecture Notes of the CIME Summer school (2005) available on line at http://cvgmt.sns.it/people/ambrosio/. | Zbl 1159.35041
,[4] Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). | MR 1857292 | Zbl 0957.49001
, and ,[5] Gradient flows in metric spaces and in the Wasserstein spaces of probability measures. Birkhäuser (2005). | MR 2129498 | Zbl 1090.35002
, and ,[6] On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Diff. Eq. 26 (2001) 43-100. | MR 1842428 | Zbl 0982.35113
, , and ,[7] A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | MR 1738163 | Zbl 0968.76069
and ,[8] Constrained steepest descent in the 2-Wasserstein metric. Ann. Math. 157 (2003) 807-846. | MR 1983782 | Zbl 1038.49040
and ,[9] Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric. Arch. Rational Mech. Anal. 172 (2004) 21-64. | MR 2048566 | Zbl 1182.76944
and ,[10] Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1-82. | MR 1853037 | Zbl 0984.35027
, , , and ,[11] Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003) 971-1018. | MR 2053570 | Zbl 1073.35127
, and ,[12] Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal. 179 (2006) 217-263. | MR 2209130 | Zbl 1082.76105
, and ,[13] The mathematics of diffusion. Clarendon Press, Oxford, second edition (1975). | MR 359551 | Zbl 0071.41401
,[14] Intrinsic distance on a Lipschitz Riemannian manifold. Rend. Sem. Mat. Univ. Politec. Torino 46 (1990) 157-170. | MR 1084564 | Zbl 0719.53047
and ,[15] Intrinsic distance on a LIP Finslerian manifold. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 17 (1993) 129-151. | MR 1268628 | Zbl 0852.53050
and ,[16] LIP manifolds: from metric to Finslerian structure. Math. Z. 218 (1995) 223-237. | MR 1318157 | Zbl 0819.53014
and ,[17] New problems on minimizing movements, in Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math. 29, Masson, Paris (1993) 81-98. | MR 1260440 | Zbl 0851.35052
,[18] Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980) 180-187. | MR 636814 | Zbl 0465.47041
, and ,[19] Evolution equations with lack of convexity. Nonlinear Anal. 9 (1985) 1401-1443. | MR 820649 | Zbl 0545.46029
, and ,[20] Probabilities and potential. North-Holland Publishing Co., Amsterdam (1978). | MR 521810 | Zbl 0494.60001
and ,[21] Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR 1158660 | Zbl 0804.28001
and ,[22] The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17 (electronic). | MR 1617171 | Zbl 0915.35120
, and ,[23] Transport via mass transportation. Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 311-338. | MR 2176295 | Zbl 1100.35028
and ,[24] Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differential Equations 28 (2007) 85-120. | MR 2267755 | Zbl 1132.60004
,[25] A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. | MR 1451422 | Zbl 0901.49012
,[26] Doubly degenerate diffusion equations as steepest descent. Manuscript (1996) available on line at http://www-mathphys.iam.uni-bonn.de/web/forschung/publikationen/main-en.htm.
,[27] Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. 52 (1999) 873-915. | MR 1682800 | Zbl 0929.76136
,[28] The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq. 26 (2001) 101-174. | MR 1842429 | Zbl 0984.35089
,[29] Variational principle for general diffusion problems. Appl. Math. Optim. 50 (2004) 229-257. | MR 2091124 | Zbl 1074.35059
and ,[30] Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl. 84 (2005) 149-168. | MR 2118836 | Zbl pre02164960
,[31] The porous medium equation, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford (2007). | MR 2286292 | Zbl 1107.35003
,[32] Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003). | MR 1964483 | Zbl 1106.90001
,[33] Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math. 58 (2005) 923-940. | MR 2142879 | Zbl 1078.53028
and ,