Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners
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| Format: | Preprint |
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2025
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| _version_ | 1866911273119121408 |
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| author | Guo, Wen Wang, Ya-Guang |
| author_facet | Guo, Wen Wang, Ya-Guang |
| contents | In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners yields intrinsic analytic obstacles: the lack of smooth tangential vectors to the boundary prevents the use of classical anisotropic Sobolev spaces, and due to the coupling of normal derivatives near corners, one can not follow the usual way to estimate the normal derivatives of solutions from the equations. To overcome these difficulties, a new class of anisotropic Sobolev spaces $H^m_*(Ω)$ is introduced to treat corner geometries. Within this framework, the well-posedness theory is obtained for both linear and nonlinear problems of the compressible ideal MHD equations with the impermeable and perfectly conducting boundary conditions. The associated linearized problem is studied in several steps: first one deduces the existence of weak solutions by using a duality argument in high order tangential spaces, then verifies that it is indeed a strong solution by several smoothing procedures preserving traces to get a weak-strong uniqueness result, afterwards the estimates of normal derivatives are obtained by combining the structure of MHD equations with the Helmholtz-type decomposition for both of velocity and magnetic fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05343 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners Guo, Wen Wang, Ya-Guang Analysis of PDEs Mathematical Physics 35L04, 76W05, 35A01 In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners yields intrinsic analytic obstacles: the lack of smooth tangential vectors to the boundary prevents the use of classical anisotropic Sobolev spaces, and due to the coupling of normal derivatives near corners, one can not follow the usual way to estimate the normal derivatives of solutions from the equations. To overcome these difficulties, a new class of anisotropic Sobolev spaces $H^m_*(Ω)$ is introduced to treat corner geometries. Within this framework, the well-posedness theory is obtained for both linear and nonlinear problems of the compressible ideal MHD equations with the impermeable and perfectly conducting boundary conditions. The associated linearized problem is studied in several steps: first one deduces the existence of weak solutions by using a duality argument in high order tangential spaces, then verifies that it is indeed a strong solution by several smoothing procedures preserving traces to get a weak-strong uniqueness result, afterwards the estimates of normal derivatives are obtained by combining the structure of MHD equations with the Helmholtz-type decomposition for both of velocity and magnetic fields. |
| title | Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners |
| topic | Analysis of PDEs Mathematical Physics 35L04, 76W05, 35A01 |
| url | https://arxiv.org/abs/2511.05343 |