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

Michel Talagrand

Martial Agueh ¹

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

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

Abstract (Original language)
Résumé

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

is

ℝ^{n}

, or a bounded domain of

ℝ^{n}

in which case

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

on

(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.

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.

Article information
Cite this article

Received: 2003-07-04
Accepted: 2003-07-17
Published online: 2003-08-22

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

Author's affiliations:

Martial Agueh ¹

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

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

@article{CRMATH_2003__337_5_331_0,
     author = {Martial Agueh},
     title = {Asymptotic behavior for doubly degenerate parabolic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {331--336},
     year = {2003},
     publisher = {Elsevier},
     volume = {337},
     number = {5},
     doi = {10.1016/S1631-073X(03)00352-2},
     language = {en},
}

TY  - JOUR
AU  - Martial Agueh
TI  - Asymptotic behavior for doubly degenerate parabolic equations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 331
EP  - 336
VL  - 337
IS  - 5
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00352-2
LA  - en
ID  - CRMATH_2003__337_5_331_0
ER  -

%0 Journal Article
%A Martial Agueh
%T Asymptotic behavior for doubly degenerate parabolic equations
%J Comptes Rendus. Mathématique
%D 2003
%P 331-336
%V 337
%N 5
%I Elsevier
%R 10.1016/S1631-073X(03)00352-2
%G en
%F CRMATH_2003__337_5_331_0

References
Cited by

[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

Cited by Sources:

Comments - Policy