Well-posedness of the relaxed Electron MHD equations with random diffusion
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| Format: | Preprint |
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2025
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| _version_ | 1866914115894640640 |
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| author | Hu, Ruimeng Peng, Qirui Yang, Xu |
| author_facet | Hu, Ruimeng Peng, Qirui Yang, Xu |
| contents | We study the three-dimensional Electron Magnetohydrodynamics (EMHD) equations without resistivity, a regime known to be ill-posed in Sobolev and Gevrey spaces due to the quasilinear nature of the system. Motivated by recent work on stochastic regularization of the inviscid primitive equations [R. Hu, Q. Lin, and R. Liu, J. Nonlinear Sci. 35:84 (2025)], we introduce a modified EMHD model where resistivity is replaced by multiplicative noise and the nonlinear term is regularized by a fractional derivative. In particular, the classical advection term $(B \cdot \nabla)J$ is replaced by its fractional version $(B \cdot \nabla^α)J$ with $0 < α\leq 1$. We show that for $α< 1$, the system is locally well-posed almost surely in suitable Gevrey spaces, and globally well-posed with high probability for small initial data. The results demonstrate that stochastic perturbations can restore well-posedness in a broader class of quasilinear magnetic models relevant to plasma dynamics and turbulence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of the relaxed Electron MHD equations with random diffusion Hu, Ruimeng Peng, Qirui Yang, Xu Analysis of PDEs Probability We study the three-dimensional Electron Magnetohydrodynamics (EMHD) equations without resistivity, a regime known to be ill-posed in Sobolev and Gevrey spaces due to the quasilinear nature of the system. Motivated by recent work on stochastic regularization of the inviscid primitive equations [R. Hu, Q. Lin, and R. Liu, J. Nonlinear Sci. 35:84 (2025)], we introduce a modified EMHD model where resistivity is replaced by multiplicative noise and the nonlinear term is regularized by a fractional derivative. In particular, the classical advection term $(B \cdot \nabla)J$ is replaced by its fractional version $(B \cdot \nabla^α)J$ with $0 < α\leq 1$. We show that for $α< 1$, the system is locally well-posed almost surely in suitable Gevrey spaces, and globally well-posed with high probability for small initial data. The results demonstrate that stochastic perturbations can restore well-posedness in a broader class of quasilinear magnetic models relevant to plasma dynamics and turbulence. |
| title | Well-posedness of the relaxed Electron MHD equations with random diffusion |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2509.18640 |