Calculus of Variations and Geometric Measure Theory

S. Lisini

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

created by lisini on 21 Feb 2008
modified on 31 Jul 2010


Published Paper

Inserted: 21 feb 2008
Last Updated: 31 jul 2010

Journal: ESAIM Control Optim. Calc. Var.
Volume: 15
Pages: 712-740
Year: 2009


We study existence and approximation of non-negative solutions of a class of nonlinear diffusion equations with variable coefficients. The results are obtained interpreting this kind of equations as ``gradient flow'' of a suitable energy functional with respect to a suitable Wasserstein distance. More precisely the Wasserstein distance between probability measures on the euclidean space endowed with the Riemannian distance induced by the inverse matrix of the coefficients of the equation. Long time asymptotic behavior and rate decay to stationary state for solutions of the equation are studied. A contraction property in Wasserstein distance for solutions of the equation is studied in a particular case.