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

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 Ω =T× [-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 Vᵢₙ V in and the initial magnetic field Bᵢₙ B in satisfy ( Vᵢₙ-(y,0), Bᵢₙ-(1,0)) _Hx,y²≤ ε min\ν ,μ \¹/2 ‖ V in - ( y , 0 ) , B in - ( 1 , 0 ) ‖ H x , y 2 ≤ ϵ min { ν , μ } 1 2 for some small ε ϵ independent of ν ,μ ν , μ , then the solution of the system remains within O(min\ν ,μ \¹/2) O ( min { ν , μ } 1 2 ) of Couette flow, and close to Couette flow as t→ ∞ t → ∞ ; the magnetic field remains within O(min\ν ,μ \¹/2) O ( min { ν , μ } 1 2 ) of the (1, 0), and close to (1, 0) as t→ ∞ t → ∞ .