Well-posedness of Wasserstein Gradient Flow Solutions of Higher Order Evolution Equations

TL;DR

This paper introduces a relaxed displacement convexity concept to prove short-term existence and uniqueness of Wasserstein gradient flows for higher order energies, applying it to equations like thin-film and quantum drift diffusion.

Contribution

It develops a new relaxed convexity framework enabling well-posedness results for complex higher order evolution equations in Wasserstein spaces.

Findings

Established short-time existence and uniqueness of solutions

Proved local and global well-posedness for specific equations

Applied theory to thin-film and quantum drift diffusion equations

Abstract

A relaxed notion of displacement convexity is defined and used to establish short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.