Asymptotic behavior for doubly degenerate parabolic equations

Agueh, Martial

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

Nous utilisons des inégalités de transport de masse pour étudier le comportement asymptotique des équations paraboliques doublement dégénérées de la forme (1), où $\Omega$ est soit ${ℝ}^n$, ou un domaine borné de ${ℝ}^n$ auquel cas $\rho \nabla c^{*}\left[\nabla (F\text{'}\left(\rho \right)+V)\right]·\nu =0$ sur $(0,\infty )\times \partial \Omega$. 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

$$\frac{\partial \rho }{\partial t}=\mathrm{div}\left\{\rho \nabla c^{*}\left[\nabla \left(F\text{'}\left(\rho \right)+V\right)\right]\right\}\mathrm{in}(0,\infty )\times \Omega ,\mathrm{and}\rho (t=0)={\rho }_0\mathrm{in}\left\{0\right\}\times \Omega ,$$

(1)

where $\Omega$ is ${ℝ}^n$, or a bounded domain of ${ℝ}^n$ in which case $\rho \nabla c^{*}\left[\nabla (F\text{'}\left(\rho \right)+V)\right]·\nu =0$ on $(0,\infty )\times \partial \Omega$. 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.

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