On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911769736249344 |
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| author | Deng, Xuemin Xiao, Yuelong Zang, Aibin |
| author_facet | Deng, Xuemin Xiao, Yuelong Zang, Aibin |
| contents | A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain $Ω=\mathbb{T}^2\times\mathbb{R}$ with $\mathbb{T}^2=[0, 1]^2$. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution $(u,b)$ into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in $H^2$, and the solution converges to its zeroth horizontal mode. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state Deng, Xuemin Xiao, Yuelong Zang, Aibin Analysis of PDEs A main result of this paper establishes the global stability of the 3D MHD equations with mixed partial dissipation near a background magnetic field in the domain $Ω=\mathbb{T}^2\times\mathbb{R}$ with $\mathbb{T}^2=[0, 1]^2$. More precisely, each velocity equation lacks its own directional dissipation, and the magnetic equation lacks vertical dissipation in the MHD system. The key point to obtain the stability result is that we decompose the solution $(u,b)$ into the zeroth horizontal mode and the non-zeroth modes and complete the desired bound with the strong Poincaré type inequalities in the treatment of several nonlinear terms. Then we focus on the large-time behavior of the solution, where the non-zeroth modes decay exponentially in $H^2$, and the solution converges to its zeroth horizontal mode. |
| title | On the stability and exponential decay of the 3D MHD system with mixed partial dissipation near a equilibrium state |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.01406 |