The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914031059599360 |
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| author | Wang, Fei Xu, Lingda Zhang, Zeren |
| author_facet | Wang, Fei Xu, Lingda Zhang, Zeren |
| contents | We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $α(σ,0,1)$ with $σ\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $γ=1$ for $σ\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $γ= 4/3$ for a general $σ\in\mathbb{R}$. In the present paper, we obtain the threshold $γ=1$ in $H^N(N>13/2)$, hence improving the above results when $σ$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $ν^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field Wang, Fei Xu, Lingda Zhang, Zeren Analysis of PDEs We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $α(σ,0,1)$ with $σ\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $γ=1$ for $σ\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $γ= 4/3$ for a general $σ\in\mathbb{R}$. In the present paper, we obtain the threshold $γ=1$ in $H^N(N>13/2)$, hence improving the above results when $σ$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $ν^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}. |
| title | The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.19822 |