Transport distances and geodesic convexity for systems of degenerate diffusion equations

TL;DR

This paper introduces Wasserstein-like transport distances for vector-valued densities, enabling the variational analysis of certain nonlinear parabolic systems and establishing conditions for geodesic convexity and existence of weak solutions.

Contribution

It develops a new class of transport distances with a mobility matrix, analyzes their properties, and applies them to prove existence of solutions for degenerate diffusion systems.

Findings

Established geodesic completeness of the new distances.

Derived a generalized McCann condition for convexity.

Proved existence of weak solutions using minimizing movement scheme.

Abstract

We introduce Wasserstein-like dynamical transport distances between vector-valued densities on the real line. The mobility function from the scalar theory is replaced by a mobility matrix, that is subject to positivity and concavity conditions. Our primary motivation is to cast certain systems of nonlinear parabolic evolution equations into the variational framework of gradient flows. In the first part of the paper, we investigate the structural properties of the new class of distances like geodesic completeness. The second part is devoted to the identification of $\lambda$-geodesically convex functionals and their $\lambda$-contractive gradient flows. One of our results is a generalized McCann condition for geodesic convexity of the internal energy. In the third part, the existence of weak solutions to a certain class of degenerate diffusion systems with drift is shown. Even if the underlying energy function is not geodesically convex w.r.t. our new distance, the construction of a weak solution is still possible using de Giorgi's minimizing movement scheme.