A well-posedness result for the compressible two-fluid model with density-dependent viscosity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916743189889024 |
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| author | Zodji, Sagbo Marcel |
| author_facet | Zodji, Sagbo Marcel |
| contents | In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space $\mathbb R^d\;(d\in \{2,3\}$). The two phases of the mixture are separated by a $\mathscr{C}^{1+α}$-regular sharp interface $\mathcal{C}$ across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is $α$-Hölder continuous on both sides of $\mathcal{C}$. The initial velocity belongs to the Sobolev space $H^1(\mathbb R^d)$, and the divergence of the initial stress tensor belongs to $L^2(\mathbb R^d)$. The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12525 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A well-posedness result for the compressible two-fluid model with density-dependent viscosity Zodji, Sagbo Marcel Analysis of PDEs 76N10, 35A02, 76T10, 35Q30 In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space $\mathbb R^d\;(d\in \{2,3\}$). The two phases of the mixture are separated by a $\mathscr{C}^{1+α}$-regular sharp interface $\mathcal{C}$ across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is $α$-Hölder continuous on both sides of $\mathcal{C}$. The initial velocity belongs to the Sobolev space $H^1(\mathbb R^d)$, and the divergence of the initial stress tensor belongs to $L^2(\mathbb R^d)$. The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions. |
| title | A well-posedness result for the compressible two-fluid model with density-dependent viscosity |
| topic | Analysis of PDEs 76N10, 35A02, 76T10, 35Q30 |
| url | https://arxiv.org/abs/2310.12525 |