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

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.