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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2506.08394 |
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| _version_ | 1866912422433914880 |
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| author | Abe, Ken Jeong, In-Jee Pasqualotto, Federico Sato, Naoki |
| author_facet | Abe, Ken Jeong, In-Jee Pasqualotto, Federico Sato, Naoki |
| contents | We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity $κ>0$ and a force proportional to $\sqrtκ\ $ on the flat $d$-torus $\mathbb{T}^{d}$ for $d\geq 2$. We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium $B(x)$ in $H^{1}(\mathbb{T}^{d})$ with law $D(B)=μ$ as a non-resistive limit $κ\to 0$ of statistically stationary solutions $B_κ(x,t)$. For $d=2$, the measure $μ$ does not concentrate on any compact sets in $H^{1}(\mathbb{T}^{2})$ with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium $B(x)$ are almost surely not finite Fourier mode solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations Abe, Ken Jeong, In-Jee Pasqualotto, Federico Sato, Naoki Analysis of PDEs We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity $κ>0$ and a force proportional to $\sqrtκ\ $ on the flat $d$-torus $\mathbb{T}^{d}$ for $d\geq 2$. We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium $B(x)$ in $H^{1}(\mathbb{T}^{d})$ with law $D(B)=μ$ as a non-resistive limit $κ\to 0$ of statistically stationary solutions $B_κ(x,t)$. For $d=2$, the measure $μ$ does not concentrate on any compact sets in $H^{1}(\mathbb{T}^{2})$ with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium $B(x)$ are almost surely not finite Fourier mode solutions. |
| title | MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.08394 |