Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type

TL;DR

This paper proves the existence and analyzes the long-term behavior of weak solutions for a class of coupled fourth-order degenerate parabolic systems modeled as gradient flows, using variational methods and Wasserstein-like metrics.

Contribution

It introduces a novel approach to establish weak solutions for complex fourth-order systems via a Wasserstein-like metric and variational minimizing movement scheme.

Findings

Existence of weak solutions under convexity assumptions

Long-time behavior characterized by variational estimates

Regularity results derived from variational methods

Abstract

This article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth-order degenerate parabolic equations of gradient flow type. The underlying metric is a Wasserstein-like transportation distance for vector-valued functions, with nonlinear mobilities in each component. Under the hypothesis of (flat) convexity of the driving free energy functional, weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows. The essential regularity estimates are derived by variational methods.