Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916033861779456 |
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| author | Davoli, Elisa Marino, Greta Pietschmann, Jan-Frederik |
| author_facet | Davoli, Elisa Marino, Greta Pietschmann, Jan-Frederik |
| contents | We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to possible degeneracies and prove, as our first main result, its global-in-time existence. The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired nonlocal weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment.
Finally, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms Davoli, Elisa Marino, Greta Pietschmann, Jan-Frederik Analysis of PDEs We study a nonlocal Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects and degenerate mobility. The nonlocality is described by means of a symmetric singular kernel. We define a notion of weak solution adapted to possible degeneracies and prove, as our first main result, its global-in-time existence. The proof relies on an application of the formal gradient flow structure of the system (to overcome the lack of a-priori estimates), combined with an extension of the boundedness-by-entropy method, in turn involving a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired nonlocal weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. Finally, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts. |
| title | Existence and local asymptotics for a system of cross-diffusion equations with nonlocal Cahn-Hilliard terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.07396 |