On steady solutions of the Hall-MHD system in Besov spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911827859865600 |
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| author | Tan, Jin Tsurumi, Hiroyuki Zhang, Xin |
| author_facet | Tan, Jin Tsurumi, Hiroyuki Zhang, Xin |
| contents | In this paper, we investigate the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in $\mathbb{R}^3.$ We first show the existence and uniqueness of solutions provided with the forces in $\dot B^{3/p-3}_{p,r}(\mathbb{R}^3)$ for $1\leq p <3$ and $r=1$. Moreover, this result can be extended to any $1\leq r\leq \infty$ whenever $p=2,$ without any additional assumption on the physical parameters. On the other hand, we establish some ill-posedness results for Hall-MHD system by using the discontinuity of the solution mapping of the three-dimensional stationary Navier-Stokes equations in \emph{critical} function spaces $\dot{B}^{3/p-1}_{p,r}(\mathbb{R}^3)$ ($p\geq 3$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03402 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On steady solutions of the Hall-MHD system in Besov spaces Tan, Jin Tsurumi, Hiroyuki Zhang, Xin Analysis of PDEs In this paper, we investigate the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in $\mathbb{R}^3.$ We first show the existence and uniqueness of solutions provided with the forces in $\dot B^{3/p-3}_{p,r}(\mathbb{R}^3)$ for $1\leq p <3$ and $r=1$. Moreover, this result can be extended to any $1\leq r\leq \infty$ whenever $p=2,$ without any additional assumption on the physical parameters. On the other hand, we establish some ill-posedness results for Hall-MHD system by using the discontinuity of the solution mapping of the three-dimensional stationary Navier-Stokes equations in \emph{critical} function spaces $\dot{B}^{3/p-1}_{p,r}(\mathbb{R}^3)$ ($p\geq 3$). |
| title | On steady solutions of the Hall-MHD system in Besov spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.03402 |