Existence of weak solutions to a class of fourth order partial differential equations with Wasserstein gradient structure

TL;DR

This paper establishes the global existence of nonnegative weak solutions for a class of fourth order PDEs with Wasserstein gradient structure, using a variational approach and entropy methods, applicable to models like thin film equations.

Contribution

It proves the existence of solutions for complex fourth order PDEs with Wasserstein gradient flow structure, extending previous results to broader classes of equations.

Findings

Existence of global weak solutions for the PDE class.

Application of entropy-dissipation methods for compactness.

Relevance to thin film and Derrida-Lebowitz-Speer-Spohn equations.

Abstract

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the $L^2$-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.