Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Issue		ESAIM: COCV Volume 15, Number 3, July-September 2009


Page(s)		712 - 740
DOI		10.1051/cocv:2008044
Published online		19 July 2008

ESAIM: COCV 15 (2009) 712-740
DOI: 10.1051/cocv:2008044

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Stefano Lisini

Dipartimento di Scienze e Tecnologie Avanzate, Università degli Studi del Piemonte Orientale, Italy. stefano.lisini@unipv.it

Received May 15, 2007. Revised February 4, 2008. Published online July 19, 2008.

Abstract
We study existence and approximation of non-negative solutions of partial differential equations of the type  

$\begin{displaymath}\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R} ^n,\qquad\qquad (0.1)\end{displaymath}$

where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty) \rightarrow[0,+\infty)$ is a suitable non decreasing function, $V:\mathbb{R} ^n \rightarrow\mathbb{R}$ is a convex function. Introducing the energy functional $\phi(u)=\int_{\mathbb{R} ^n} F(u(x))\,{\rm d}x+\int_{\mathbb{R} ^n}V(x)u(x)\,{\rm d}x$ , where F is a convex function linked to f by f(u) = uF'(u)-F(u), we show that u is the “gradient flow” of $\phi$ with respect to the 2-Wasserstein distance between probability measures on the space $\mathbb{R} ^n$ , endowed with the Riemannian distance induced by A^-1. In the case of uniform convexity of V, long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.

Mathematics Subject Classification. 35K55, 35K15, 35B40

Key words: Nonlinear diffusion equations, parabolic equations, variable coefficient parabolic equations, gradient flows, Wasserstein distance, asymptotic behaviour

© EDP Sciences, SMAI 2008

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