Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows
Authors
- Gau, Moritz Immanuel
- Hopf, Katharina
ORCID: 0000-0002-6527-2256
2020 Mathematics Subject Classification
- 35M33 35G50 35A15 47H05 35B40 35Q92 76A10
Keywords
- Cahn--Hilliard equation, viscoelastic phase separation, gradient systems, Cauchy problem, convexity, uniqueness, minimising movements, relaxation limit
DOI
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.
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