Nonlinear mobility continuity equations and generalized displacement convexity

TL;DR

This paper explores the geometry of measure spaces with generalized nonlinear mobilities, providing conditions for convexity, and analyzing energy landscapes, with implications for nonlinear diffusion equations.

Contribution

It introduces a generalized framework for the Wasserstein distance with nonlinear mobilities, extending convexity conditions and analyzing energy landscapes without requiring global geodesic information.

Findings

Explicit convexity condition for internal energy generalizing McCann's criterion

Existence and stability results for nonlinear diffusion equations

Potential and interaction energies are likely not displacement semiconvex

Abstract

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.