Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction

François Bolley (1) , José A. Carrillo (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics [Imperial College London]

It is well known that nonlinear diffusion equations can be interpreted as a gradient flow in the space of probability measures equipped with the Euclidean Wasserstein distance. Under suitable convexity conditions on the nonlinearity, due to R. J. McCann, the associated entropy is geodesically convex, which implies a contraction type property between all solutions with respect to this distance. In this note, we give a simple straightforward proof of the equivalence between this contraction type property and this convexity condition, without even resorting to the entropy and the gradient flow structure.

Domaines: Equations aux dérivées partielles [math.AP]

François Bolley, José A. Carrillo. Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction. 2013. ⟨hal-00859401v2⟩