On axisymmetric self-similar solutions to the MHD system
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910053408178176 |
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| author | Zhang, Shaoheng |
| author_facet | Zhang, Shaoheng |
| contents | Let $(\mathbf{u},\mathbf{B})$ be an axisymmetric self-similar solution to the stationary MHD equations with magnetic diffusion, of the form $\mathbf{u}=u^r(r,z)\mathbf{e}_{r}+u^θ(r,z)\mathbf{e}_θ+u^z(r,z)\mathbf{e}_{z}$ and $\mathbf{B}=B^θ(r,z)\mathbf{e}_θ$ in cylindrical coordinates $(r,θ,z)$, where $(\mathbf{e}_r,\mathbf{e}_θ,\mathbf{e}_z)$ is the orthonormal basis. Under the assumption that $u^r < \frac{1}{3r} + \frac{2r}{3}$ on the unit sphere and on its intersection with the half-space, respectively, we prove two main results. First, for the domain $\mathbb{R}^3\setminus\{0\}$, the velocity field $\mathbf{u}$ must be a Landau solution and the magnetic field $\mathbf{B} \equiv 0$. Second, in the half-space $\mathbb{R}^3_+$ with either the no-slip or Navier slip boundary condition, we establish that all such axisymmetric self-similar solutions are trivial, i.\,e., $\mathbf{u}=\mathbf{B}=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21194 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On axisymmetric self-similar solutions to the MHD system Zhang, Shaoheng Analysis of PDEs Let $(\mathbf{u},\mathbf{B})$ be an axisymmetric self-similar solution to the stationary MHD equations with magnetic diffusion, of the form $\mathbf{u}=u^r(r,z)\mathbf{e}_{r}+u^θ(r,z)\mathbf{e}_θ+u^z(r,z)\mathbf{e}_{z}$ and $\mathbf{B}=B^θ(r,z)\mathbf{e}_θ$ in cylindrical coordinates $(r,θ,z)$, where $(\mathbf{e}_r,\mathbf{e}_θ,\mathbf{e}_z)$ is the orthonormal basis. Under the assumption that $u^r < \frac{1}{3r} + \frac{2r}{3}$ on the unit sphere and on its intersection with the half-space, respectively, we prove two main results. First, for the domain $\mathbb{R}^3\setminus\{0\}$, the velocity field $\mathbf{u}$ must be a Landau solution and the magnetic field $\mathbf{B} \equiv 0$. Second, in the half-space $\mathbb{R}^3_+$ with either the no-slip or Navier slip boundary condition, we establish that all such axisymmetric self-similar solutions are trivial, i.\,e., $\mathbf{u}=\mathbf{B}=0$. |
| title | On axisymmetric self-similar solutions to the MHD system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.21194 |