Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913522546376704 |
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| author | Jing, Tian Wang, Dehua |
| author_facet | Jing, Tian Wang, Dehua |
| contents | In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain $Ω\subseteq \mathbb{R}^3$. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fréchet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19511 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations Jing, Tian Wang, Dehua Analysis of PDEs 35Q30, 35Q35, 76D05, 76W05, 76D27, 76Txx In this paper, we study the existence of strong solutions to the two-phase magnetohydrodynamic equations in a bounded domain $Ω\subseteq \mathbb{R}^3$. The fluids are incompressible, viscous, and resistive. The surface tension is considered. The equations are reformulated using the Hanzawa transformation, which turns the free interface into a fixed one for a short time. The study of the new equations is then divided into the principal part and the nonlinear part. Due to the effect of the magnetic field and the complexity of the transformation in generic bounded domains, the Fréchet derivatives of nonlinearities have to be carefully estimated. The equations can then be solved using the fixed-point argument by finding a contraction mapping, which follows the estimates of the nonlinear part. |
| title | Strong solutions to the three-dimensional two-phase magnetohydrodynamic equations |
| topic | Analysis of PDEs 35Q30, 35Q35, 76D05, 76W05, 76D27, 76Txx |
| url | https://arxiv.org/abs/2409.19511 |