Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential
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| Format: | Preprint |
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2025
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| _version_ | 1866910011921268736 |
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| author | Grasselli, Maurizio Poiatti, Andrea |
| author_facet | Grasselli, Maurizio Poiatti, Andrea |
| contents | We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_17296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential Grasselli, Maurizio Poiatti, Andrea Analysis of PDEs 35B36, 35B40, 35D30, 35Q35, 76T99 We consider the classical initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn--Hilliard-Navier--Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely. |
| title | Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential |
| topic | Analysis of PDEs 35B36, 35B40, 35D30, 35Q35, 76T99 |
| url | https://arxiv.org/abs/2510.17296 |