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

Abstract

We study existence and approximation of non-negative solutions of partial differential equations of the type where is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, is a suitable non decreasing function, is a convex function. Introducing the energy functional, where is a convex function linked to by, we show that is the "gradient flow" of with respect to the 2-Wasserstein distance between probability measures on the space, endowed with the Riemannian distance induced by In the case of uniform convexity of, 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. © EDP Sciences, SMAI 2008.