Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915618924527616 |
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| author | Abels, Helmut Garcke, Harald Poiatti, Andrea |
| author_facet | Abels, Helmut Garcke, Harald Poiatti, Andrea |
| contents | A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness Abels, Helmut Garcke, Harald Poiatti, Andrea Analysis of PDEs 35Q30, 76D05, 35D35, 35Q35, 76T06 A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system. |
| title | Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness |
| topic | Analysis of PDEs 35Q30, 76D05, 35D35, 35Q35, 76T06 |
| url | https://arxiv.org/abs/2407.14941 |