Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 712-740.

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

t u-div(A((f(u))+uV))=0in(0,+)× n ,(0.1)
where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, f:[0,+)[0,+) is a suitable non decreasing function, V: n is a convex function. Introducing the energy functional φ(u)= n F(u(x))dx+ n V(x)u(x)dx, 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 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.

DOI : 10.1051/cocv:2008044
Classification : 35K55, 35K15, 35B40
Mots clés : nonlinear diffusion equations, parabolic equations, variable coefficient parabolic equations, gradient flows, Wasserstein distance, asymptotic behaviour
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     title = {Nonlinear diffusion equations with variable coefficients as gradient flows in {Wasserstein} spaces},
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Lisini, Stefano. Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 712-740. doi : 10.1051/cocv:2008044. http://www.numdam.org/articles/10.1051/cocv:2008044/

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