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

Abstract

AbstractIn 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 → ∞ .