Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows

TL;DR

This paper develops a gradient-flow framework for a viscoelastic phase separation model, proving well-posedness and analyzing asymptotic limits, thereby unifying several classical phase separation equations.

Contribution

It introduces a novel gradient-flow approach to a viscoelastic phase separation model, establishing global well-posedness and stability for a broad class of initial data.

Findings

Proved global well-posedness for the model.

Established existence of gradient-flow solutions for finite-energy data.

Derived asymptotic limits recovering classical phase separation equations.

Abstract

This article is concerned with a gradient-flow approach to a Cahn-Hilliard model for viscoelastic phase separation introduced by Zhou et al. (Phys. Rev. E, 2006) in its variant with constant mobility. By means of time-incremental minimisation and generalised contractivity estimates, we establish the global well-posedness of the Cauchy problem for moderately regular initial data. For general finite-energy data we obtain the existence of gradient-flow solutions and a stability estimate of weak-strong type. We further study the asymptotic behaviour for relaxation time and bulk modulus depending on a small parameter. Depending on the scaling, we recover the Cahn-Hilliard, the mass-conserving Allen-Cahn or the viscous Cahn-Hilliard equation. A challenge in the well-posedness analysis is the failure of semiconvexity of the appropriate driving functional, which is caused by a phase-dependence of the bulk modulus.