Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918128493002752 |
|---|---|
| author | Gau, Moritz Immanuel Hopf, Katharina |
| author_facet | Gau, Moritz Immanuel Hopf, Katharina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows Gau, Moritz Immanuel Hopf, Katharina Analysis of PDEs 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. |
| title | Well-posedness and relaxation in a simplified model for viscoelastic phase separation via Hilbertian gradient flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.10722 |