Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, p. 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 : https://doi.org/10.1051/cocv:2008044
Classification:  35K55,  35K15,  35B40
Keywords: nonlinear diffusion equations, parabolic equations, variable coefficient parabolic equations, gradient flows, Wasserstein distance, asymptotic behaviour
@article{COCV_2009__15_3_712_0,
     author = {Lisini, Stefano},
     title = {Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     pages = {712-740},
     doi = {10.1051/cocv:2008044},
     zbl = {1178.35201},
     mrnumber = {2542579},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_3_712_0}
}
Lisini, Stefano. Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 3, pp. 712-740. doi : 10.1051/cocv:2008044. http://www.numdam.org/item/COCV_2009__15_3_712_0/

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