Contractivity and Asymptotics in Wasserstein Metrics for Viscous Nonlinear Scalar Conservation Laws

Carrillo, José A. ; Di Francesco, Marco ; Lattanzio, Corrado

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 277-292 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

In this work, recent results concerning the long time asymptotics of one- dimensional scalar conservation laws with probability densities as initial data are reviewed and further applied to the case of viscous conservation laws with nonlinear degenerate diffusions. The non-strict contraction of the maximal transport distance together with a uniform expansion of the solutions lead to the existence of time-de- pendent asymptotic profiles for a large class of convection-diffusion problems with fully general nonlinearities and with degenerate diffusion.

Résumé

In questo articolo sono riportati alcuni risultati recenti riguardo il comportamento asintotico nel tempo di leggi di conservazione scalari in una dimensione spaziale e con densità di probabilità come dati iniziali. Tali risultati sono quindi applicati al caso di leggi di conservazione viscose con diffusioni nonlineari degeneri. Le proprietà di contrazione nella distanza di trasporto massimale e di uniforme espansione delle soluzioni forniscono l'esistenza di profili asintotici dipendenti dal tempo per un'ampia classe di equazioni di convenzione-diffusione con nonlinearità arbitrarie e diffusione degenere.

Détail

Publié le : 2007-06-01

MR 2339442 | Zbl 1178.35390

Bibliographie

[1] Agueh, M., 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] Ambrosio, L. - Gigli, N. - Savaré, G., Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 2005. | MR 2129498

[3] Andreucci, D., Degenerate parabolic equations with initial data measures, Trans. Amer. Math. Soc., 349 (10) (1997), 3911-3923. | MR 1333384 | Zbl 0885.35056

[4] Bénilan, P. - Bouillet, J. E., On a parabolic equation with slow and fast diffusions, Nonlinear Anal., 26 (4) (1996), 813-822. | MR 1362754 | Zbl 0840.35052

[5] Bolley, F. - Brenier, Y. - Loeper, G., Contractive metrics for scalar conservation laws, J. Hyperbolic Differ. Equ., 2 (2005), 91-107. | MR 2134955 | Zbl 1071.35081

[6] Bolley, F. - Carrillo, J. A., Tanaka Theorem for Inelastic Maxwell Models, To appear in Comm. Math. Phys. | MR 2346391

[7] Brenier, Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), 411-452, 1999. | MR 1658919 | Zbl 0910.35098

[8] Brenier, Y., $L^2$ formulation of multidimensional scalar conservation laws, to appear in Arch. Ration. Mech. Anal. | MR 2506069 | Zbl 1180.35346

[9] Carlen, E. - Gangbo, W., Constrained steepest descent in the 2-Wasserstein metric, Ann. of Math., 157 (2003), 807-846. | MR 1983782 | Zbl 1038.49040

[10] Carrillo, J., Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. | MR 1709116 | Zbl 0935.35056

[11] Carrillo, J. A. - Di Francesco, M. - Gualdani, M. P., Semidiscretization and long- time asymptotics of nonlinear diffusion equations, Comm. Math. Sci., 1 (2007), 21-53. | MR 2301287 | Zbl 1145.35028

[12] Carrillo, J. A. - Di Francesco, M. - Lattanzio, C., Contractivity of Wasserstein metrics and asymptotic profiles for scalar conservation laws, J. Differential Equations, 231 (2006), 425-458. | MR 2287892 | Zbl 1168.35390

[13] Carrillo, J. A. - Di Francesco, M. - Toscani, G., Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for $u_t = \Delta \phi (u)$, Arch. Rational Mech. Anal., 180 (2006), 127-149. | MR 2211709 | Zbl 1096.35015

[14] Carrillo, J. A. - Gualdani, M. P. - Toscan, G., Finite speed of propagation in porous media by mass transportation methods, C. R. Acad. Sci. Paris, 338 (2004), 815-818. | MR 2059493

[15] Carrillo, J. A., Mccann, R. J. - Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., 179 (2006), 217-264. | MR 2209130 | Zbl 1082.76105

[16] Carrillo, J. A. - Toscani, G., Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, In New trends in mathematical physics, World Sci. Publ., Hackensack, NJ, 2004, 234-244. | MR 2163983 | Zbl 1089.76055

[17] Carrillo, J. A. - Vázquez, J. L., Asymptotic Complexity in Filtration Equations, To appear in J. Evol. Equ. | MR 2328935

[18] Crandall, M. - Pierre, M., Regularizing effects for $u_t +A\phi (u)=0$ in $L^1$, J. Funct. Anal., 45 (1982), 194-212. | MR 647071 | Zbl 0483.35076

[19] Cullen, M. - Gangbo, W., A variational approach for the 2-dimensional semi- geostrophic shallow water equations, Arch. Ration. Mech. Anal., 156 (2001) 241-273. | MR 1816477 | Zbl 0985.76008

[20] Di Francesco, M. - Markowich, P. A., Entropy dissipation and Wasserstein metric methods for the viscous Burgers' equation: convergence to diffusive waves, In Partial Differential Equations and Inverse Problems, 362 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2004, 145-165. | MR 2091496 | Zbl 1062.35111

[21] Di Francesco, M. - Wunsch, M., Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models, to appear in Monat. Math. | MR 2395521 | Zbl 1159.35008

[22] Evje, S. - Karlsen, K. H., Viscous splitting approximation of mixed hyperbolic- parabolic convection-diffusion equations, Numer. Math., 83 (1) (1999), 107-137. | MR 1702599 | Zbl 0961.65084

[23] Escobedo, M. - Vázquez, J. L. - Zuazua, E., Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1) (1993), 43-65. | MR 1233647

[24] Hopf, E., The partial differential equation $u_t + uu_x = \mu u_{xx}$, Comm. Pure Appl. Math., 3 (1950), 201-230. | MR 47234 | Zbl 0039.10403

[25] Kalashnikov, A. S., Some problems of the qualitative theory of second-order non-linear degenerate parabolic equations, Uspekhi Mat. Nauk, 42 (2(254)) (1987), 135- 176, 287. | MR 898624

[26] Laurençot, P. - Simondon, F., Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A, 128 (2) (1998), 315-336. | MR 1621331 | Zbl 0906.35050

[27] Liu, T. P. - Pierre, M., Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations, 51 (1984), 419-441. | MR 735207 | Zbl 0545.35057

[28] Lott, J. - Villani, C., Ricci curvature for metric-measure spaces via optimal transport, To appear in Ann. of Math. | MR 2480619 | Zbl 1178.53038

[29] Mccann, R. J., Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-632. | MR 2219334 | Zbl 1096.85006

[30] Otto, F., The geometry of dissipative evolution equation: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. | MR 1842429 | Zbl 0984.35089

[31] Otto, F. - Westdickenberg, M., Eulerian calculus for the contraction in the wasserstein distance, SIAM J. Math. Anal., 37 (2005), 1227-1255. | MR 2192294 | Zbl 1094.58016

[32] Sturm, K. T., Convex functionals of probability measures and nonlinear diffusions on manifolds, J. Math. Pures Appl., 84 (2005) 149-168. | MR 2118836 | Zbl 1259.49074

[33] Toscani, G., A central limit theorem for solutions of the porous medium equation, J. Evol. Equ., 5 (2005), 185-203. | MR 2133441 | Zbl 1082.35091

[34] Vázquez, J. L., The Porous Medium Equation. New contractivity results, Progress in Nonlinear Differential Equations and Their Applications, 63 (205) (Volume in honor of H. Brezis, Proceedings of Gaeta Congress, June 2004), 433-451. | MR 2176734

[35] Villani, C., Topics in optimal transportation, 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. | MR 1964483

[36] Villani, C., Optimal transport, old and new, Lecture Notes for the 2005 Saint-Flour summer school, to appear in Springer2007. | MR 2815763