Asymptotic behavior for doubly degenerate parabolic equations

Martial Agueh

doi:10.1016/S1631-073X(03)00352-2

Probability Theory/Ordinary Differential Equations

Asymptotic behavior for doubly degenerate parabolic equations
[Comportement asymptotique des équations paraboliques doublement dégénérées]

Présenté par : Michel Talagrand

Martial Agueh ¹

¹ Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 331-336.

Résumé
Abstract

Nous utilisons des inégalités de transport de masse pour étudier le comportement asymptotique des équations paraboliques doublement dégénérées de la forme (1), où $Ω$ est soit $ℝ^{n}$ , ou un domaine borné de $ℝ^{n}$ auquel cas $ρ \nabla c^{*} [\nabla (F' (ρ) + V)] \cdot ν = 0$ sur $(0, \infty) \times \partial Ω$ . Nous examinons le cas où le potentiel V est uniformément c-convexe, et le cas dégénéré où V=0. Dans ces deux cas, nous montrons une décroissance exponentielle de la différence d'entropies et de la distance de Wasserstein – suivant le coût c – des solutions de l'équation et de sa solution stationnaire, et nous précisons les taux de convergence. En particulier, nous généralisons à tous les p>1 les inégalités HWI obtenues dans Otto et Villani (J. Funct. Anal. 173 (2) (2000) 361–400) lorsque p=2. Cette classe d'équations contient les équations de Fokker–Planck, des milieux poreux et du p-Laplacien.

We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form

\frac{\partial ρ}{\partial t} = div \{ρ \nabla c^{*} [\nabla (F' (ρ) + V)]\} in (0, \infty) \times Ω, and ρ (t = 0) = ρ_{0} in {0} \times Ω,

(1)

where

Ω

ℝ^{n}

, or a bounded domain of

ℝ^{n}

in which case

ρ \nabla c^{*} [\nabla (F' (ρ) + V)] \cdot ν = 0

(0, \infty) \times \partial Ω

. We investigate the case where the potential V is uniformly c-convex, and the degenerate case where V=0. In both cases, we establish an exponential decay in relative entropy and in the c-Wasserstein distance of solutions – or self-similar solutions – of (1) to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all p>1, the HWI inequalities obtained by Otto and Villani (J. Funct. Anal. 173 (2) (2000) 361–400) when p=2. This class of PDEs includes the Fokker–Planck, the porous medium, fast diffusion and the parabolic p-Laplacian equations.

Reçu le : 2003-07-04
Accepté le : 2003-07-17
Publié le : 2003-08-22

DOI : 10.1016/S1631-073X(03)00352-2

Affiliations des auteurs :

Martial Agueh ¹

¹ Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

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     author = {Martial Agueh},
     title = {Asymptotic behavior for doubly degenerate parabolic equations},
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     doi = {10.1016/S1631-073X(03)00352-2},
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Martial Agueh. Asymptotic behavior for doubly degenerate parabolic equations. Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 331-336. doi : 10.1016/S1631-073X(03)00352-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00352-2/

Bibliographie
Cité par

[1] M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory, Preprint, 2002

[2] M. Agueh, N. Ghoussoub, X. Kang, Geometric inequalities via a general comparison principle for interacting gases, GAFA (2003), in press

[3] J.A. Carrillo; A. Jüngel; P.A. Markowich; G. Toscani; A. Unterreiter Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001) no. 1, pp. 1-82

[4] D. Cordero-Erausquin, W. Gangbo, C. Houdré, Inequalities for generalized entropy and optimal transportation, in: Proceedings of the Workshop: Mass Transportation Methods in Kinetic Theory and Hydrodynamics, in press

[5] M. Del Pino; J. Dolbeault Nonlinear diffusion and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving p-Laplacian, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 334 (2002) no. 5, pp. 365-370

[6] S. Kamin; J.L. Vázquez Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, Volume 4 (1988) no. 2, pp. 339-354

[7] F. Otto The geometry of dissipative evolution equation: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001) no. 1–2, pp. 101-174

[8] F. Otto; C. Villani Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., Volume 173 (2000) no. 2, pp. 361-400

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