Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system
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| Format: | Preprint |
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2024
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| _version_ | 1866915893735325696 |
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| author | Zaidni, Azeddine Benjelloun, Saad Boukharfane, Radouan |
| author_facet | Zaidni, Azeddine Benjelloun, Saad Boukharfane, Radouan |
| contents | We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $Ω\subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by $\mathfrak{F}= \int_Ω \fracε{2}\, Γ^2(\nabla ϕ) $. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions $(d=2,3)$. A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05757 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system Zaidni, Azeddine Benjelloun, Saad Boukharfane, Radouan Analysis of PDEs We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $Ω\subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by $\mathfrak{F}= \int_Ω \fracε{2}\, Γ^2(\nabla ϕ) $. The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in \cite{anderson2000phase,taylor-cahn98, zaidni2024}. Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions $(d=2,3)$. A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument. |
| title | Global existence of weak solutions to incompressible anisotropic Cahn-Hilliard-Navier-Stokes system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.05757 |