Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915419025047552 |
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| author | Zhang, Junyan |
| author_facet | Zhang, Junyan |
| contents | Current-vortex sheet is one of the characteristic discontinuities in ideal compressible magnetohydrodynamics (MHD). The motion of current-vortex sheets is described by a free-interface problem of two-phase MHD flows with magnetic fields tangential to the interface. This paper is the first part of the two-paper sequence, which aims to present a comprehensive study for compressible current-vortex sheets with or without surface tension. In this paper, we prove the local well-posedness and the incompressible limit of current-vortex sheets with surface tension. The key observation is a hidden structure of Lorentz force in the vorticity analysis which motivates us to establish the uniform estimates in anisotropic-type Sobolev spaces with weights of Mach number determined by the number of tangential derivatives. Besides, our framework of iteration and approximation to prove the local existence of vortex-sheet problems does not rely on Nash-Moser iteration. Furthermore, the local existence of current-vortex sheets without surface tension can be proved by taking zero-surface-tension limit under certain stability conditions, which is established in [73] (the second part of the two-paper sequence). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11254 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD Zhang, Junyan Analysis of PDEs Mathematical Physics Current-vortex sheet is one of the characteristic discontinuities in ideal compressible magnetohydrodynamics (MHD). The motion of current-vortex sheets is described by a free-interface problem of two-phase MHD flows with magnetic fields tangential to the interface. This paper is the first part of the two-paper sequence, which aims to present a comprehensive study for compressible current-vortex sheets with or without surface tension. In this paper, we prove the local well-posedness and the incompressible limit of current-vortex sheets with surface tension. The key observation is a hidden structure of Lorentz force in the vorticity analysis which motivates us to establish the uniform estimates in anisotropic-type Sobolev spaces with weights of Mach number determined by the number of tangential derivatives. Besides, our framework of iteration and approximation to prove the local existence of vortex-sheet problems does not rely on Nash-Moser iteration. Furthermore, the local existence of current-vortex sheets without surface tension can be proved by taking zero-surface-tension limit under certain stability conditions, which is established in [73] (the second part of the two-paper sequence). |
| title | Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2312.11254 |