Competing effects in fourth-order aggregation-diffusion equations

TL;DR

This paper establishes precise conditions for the global existence of solutions to a complex fourth-order gradient flow equation, analyzing the interplay of competing energies and extending results to coupled systems.

Contribution

It provides sharp criteria for global solutions of a Cahn-Hilliard-type equation with degenerate diffusion and explores a related coupled system, advancing understanding of such gradient flows.

Findings

Derived sharp existence conditions for the equation

Analyzed critical regimes via homogeneity of functionals

Extended results to a coupled system of equations

Abstract

We give sharp conditions for global in time existence of gradient flow solutions to a Cahn-Hilliard-type equation, with backwards second order degenerate diffusion, in any dimension and for general initial data. Our equation is the 2-Wasserstein gradient flow of a free energy with two competing effects: the Dirichlet energy and the power-law internal energy. Homogeneity of the functionals reveals critical regimes that we analyse. Sharp conditions for global in time solutions, constructed via the minimising movement scheme, also known as JKO scheme, are obtained. Furthermore, we study a system of two Cahn-Hilliard-type equations exhibiting an analogous gradient flow structure.