Gradient flows driven by a non-smooth repulsive interaction potential

TL;DR

This thesis studies the Wasserstein gradient flow of a non-smooth, repulsive interaction potential with a -|x| behavior near zero, establishing existence and uniqueness in one dimension despite non-convexity.

Contribution

It extends the theory of Wasserstein gradient flows to a non-smooth, non-convex potential, proving existence and uniqueness in one dimension.

Findings

Existence of solutions for the gradient flow with the non-smooth potential

Uniqueness of solutions in the one-dimensional case

Extension of Wasserstein gradient flow theory to non-convex potentials

Abstract

This thesis analyze the Wasserstein gradient flow of a functional defined as a double convolution of a non-smooth repulsive interaction potential. To be more precise, the potential under investigation has a -|x| behavior close to the origin. The already existent machinery of Wasserstein gradient flow is well posed for lambda-convex potential. In this case this property is lost, but it is proven that in the one dimensional case existence and uniqueness of the solution is still achieved.