Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity
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| Format: | Preprint |
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2024
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| _version_ | 1866914773597159424 |
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| author | Wang, Wendong Yang, Guoxu |
| author_facet | Wang, Wendong Yang, Guoxu |
| contents | In this paper, we investigate Liouville type theorems for the three-dimensional steady-state MHD or Hall-MHD system under some asymptotic assumptions at infinity. Firstly, for the Hall-MHD system we obtain that $u$ and $B$ are constant vectors for any fluid viscosity, magnetic resistivity or Hall-coefficient when the magnetic field $B$ tends to a non-zero constant vector at infinity while the velocity field $u$ tends to $0$. Secondly, it also follows that $u$ and $B$ are constant for the Hall-MHD system when the velocity field tends to a constant vector at infinity while the magnetic field tends to $0$ without any assumptions on viscosity, magnetic resistivity or Hall-coefficient. One main difficulty lies in the Hall term, and we obtain the $L^p$ estimates of a generalized Oseen system with some supercritical terms via Lizorkin's theory and prove that the operator is stable by exploring Kato's stability theorem. Moreover, some similar results for the degenerate fluid viscosity or magnetic resistivity for the MHD system are also obtained, which is independent of interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity Wang, Wendong Yang, Guoxu Analysis of PDEs In this paper, we investigate Liouville type theorems for the three-dimensional steady-state MHD or Hall-MHD system under some asymptotic assumptions at infinity. Firstly, for the Hall-MHD system we obtain that $u$ and $B$ are constant vectors for any fluid viscosity, magnetic resistivity or Hall-coefficient when the magnetic field $B$ tends to a non-zero constant vector at infinity while the velocity field $u$ tends to $0$. Secondly, it also follows that $u$ and $B$ are constant for the Hall-MHD system when the velocity field tends to a constant vector at infinity while the magnetic field tends to $0$ without any assumptions on viscosity, magnetic resistivity or Hall-coefficient. One main difficulty lies in the Hall term, and we obtain the $L^p$ estimates of a generalized Oseen system with some supercritical terms via Lizorkin's theory and prove that the operator is stable by exploring Kato's stability theorem. Moreover, some similar results for the degenerate fluid viscosity or magnetic resistivity for the MHD system are also obtained, which is independent of interest. |
| title | Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.18051 |