Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

TL;DR

This paper proves the global existence of weak solutions for a complex multicomponent Cahn-Hilliard system with cross-diffusion, degenerate mobility, and partial phase separation, using a gradient flow approach and entropy methods.

Contribution

It introduces a novel framework for establishing weak solutions in a degenerate, cross-diffusive Cahn-Hilliard model with partial phase separation.

Findings

Proved global existence of weak solutions.

Developed a new entropy-based analytical method.

Handled low regularity of Cahn-Hilliard terms.

Abstract

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment.