Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility
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| Format: | Preprint |
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2025
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| _version_ | 1866918368129318912 |
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| author | Stange, Jonas |
| author_facet | Stange, Jonas |
| contents | We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility Stange, Jonas Analysis of PDEs Mathematical Physics 35K35, 35D30, 35A01, 35A02, 35Q92, 35B65 We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest. |
| title | Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility |
| topic | Analysis of PDEs Mathematical Physics 35K35, 35D30, 35A01, 35A02, 35Q92, 35B65 |
| url | https://arxiv.org/abs/2507.16508 |