Nonlinear mobility continuity equations and generalized displacement convexity

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 nonrigorous argument indicating that they are not displacement semiconvex.

A User’s Guide to Optimal Transport

“…By the Riesz representation theorem there exists a vector field 26) and whose norm in L 2 (µ t0 ) is bounded above by the metric derivative |μ t | at t = t 0 . It remains to prove that the continuity equation is satisfied in the sense of distributions.…”

Example of a displacement convex functional of first order

“…The reader can see works of Lott [15], McCann and Topping [19], and references therein. Finally Lisini, Savaré and the authors [5] have considered an adapted notion of displacement convexity to problems where the metric is a generalization of the quadratic transportation distance with a nonlinear weight.…”

Scalar conservation laws seen as gradient flows: known results and new perspectives

“…The correct way to look at a scalar conservation law as a gradient flow is replacing the classical 2-Wasserstein distance with an alternative transport distance with a nonlinear mobility, see [15,24,42]. We shall describe such approach here.…”

“…Here X = X ρ according to the notation established in section 2. The class of distances (50) has been studied in [15,24,42], together with their relation with certain applied PDEs in which a nonlinear mobility effect is relevant in the dynamics, see e. g. [13] for chemotaxis movements. Let us now consider the functional F : M → R…”