Asymptotic stability of the 2D MHD equations without magnetic diffusion

Li-hua, Dong; Ren, Xiaoxia

doi:10.1063/5.0112577

Journal of Mathematical Physics

2023

DOI: 10.1063/5.0112577

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Asymptotic stability of the 2D MHD equations without magnetic diffusion

Dong Li-hua

Xiaoxia Ren

Abstract: We investigate the asymptotic stability of certain steady solution to the 2D MHD equations without magnetic diffusion in an infinite strip domain. Decay rates of smooth solution to that system are also given.

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2024

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Cited by 2 publications

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Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Jin,

Jiu,

Xie

2024

Z. Angew. Math. Phys.

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Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Jin,

Jiu,

Xie

2024

Z. Angew. Math. Phys.

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Asymptotic Stability of Two-Dimensional Magnetohydrodynamic System Near the Couette Flow in a Finite Channel

Luo,

Li,

2024

J Nonlinear Math Phys

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In this paper, we consider the asymptotic stability of the incompressible two-dimensional(2D) magnetohydrodynamic(MHD) system near the Couette flow at high Reynolds number and high magnetic Reynolds number in a finite channel $$\Omega =\mathbb {T}\times [-1,1]$$ Ω = T × [ - 1 , 1 ] . We extend the results of the Navier–Stokes equations (for the previous results see[10]) to the MHD system. We prove that if the initial velocity $$V_{in}$$ V in and the initial magnetic field $$B_{in}$$ B in satisfy $$\Vert \left( V_{in}-(y,0), B_{in}-(1,0)\right) \Vert _{H_{x,y}^{2}}\le \epsilon \text {min}\{\nu ,\mu \}^\frac{1}{2}$$ ‖ V in - ( y , 0 ) , B in - ( 1 , 0 ) ‖ H x , y 2 ≤ ϵ min { ν , μ } 1 2 for some small $$\epsilon$$ ϵ independent of $$\nu ,\mu$$ ν , μ , then the solution of the system remains within $$\mathcal{O}(\text {min}\{\nu ,\mu \}^\frac{1}{2})$$ O ( min { ν , μ } 1 2 ) of Couette flow, and close to Couette flow as $$t\rightarrow \infty$$ t → ∞ ; the magnetic field remains within $$\mathcal{O}(\text {min}\{\nu ,\mu \}^\frac{1}{2})$$ O ( min { ν , μ } 1 2 ) of the (1, 0), and close to (1, 0) as $$t\rightarrow \infty$$ t → ∞ .

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Asymptotic stability of the 2D MHD equations without magnetic diffusion

Abstract: We investigate the asymptotic stability of certain steady solution to the 2D MHD equations without magnetic diffusion in an infinite strip domain. Decay rates of smooth solution to that system are also given.

Cited by 2 publications

References 15 publications

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Asymptotic Stability of Two-Dimensional Magnetohydrodynamic System Near the Couette Flow in a Finite Channel

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