ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2009 - Volume 15, Issue 03  

Research Article

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

Lisini, Stefano

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

Abstract

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 $$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$ 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 ϕ 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.

(Received May 15 2007)

(Revised February 4 2008)

(Online publication July 19 2008)

Key Words:

  • Nonlinear diffusion equations;
  • parabolic equations;
  • variable coefficient parabolic equations;
  • gradient flows;
  • Wasserstein distance;
  • asymptotic behaviour

Mathematics Subject Classification:

  • 35K55;
  • 35K15;
  • 35B40