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

Stefano Lisini — 2009

ESAIM: Control, Optimisation and Calculus of Variations

We study existence and approximation of non-negative solutions of partial differential equations of the type t u - div ( A ( ( f ( u ) ) + u V ) ) = 0 in ( 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 ) ) d x + n V ( x ) u ( x ) d x , where F is a convex function linked to f by f ( u ) = u F ' ( u ) - F ( u ) , we show that u is the “gradient flow” of φ with respect to the 2-Wasserstein distance between probability...

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

Stefano Lisini — 2008

ESAIM: Control, Optimisation and Calculus of Variations

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 t u - div ( A ( ( f ( u ) ) + u V ) ) = 0 in ( 0 , + ) × n , ( 0 . 1 ) where 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 ) ) d x + n V ( x ) u ( x ) d x , where is a convex function linked to by f ( u ) = u F ' ( u ) - F ( u ) , we show that is the “gradient flow” of with respect to the 2-Wasserstein distance between probability measures on the space...

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