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

Abstract

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 \(\Omega =\mathbb {T}\times [-1,1]\). 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 \(V_{in}\) and the initial magnetic field \(B_{in}\) satisfy \(\Vert \left( V_{in}-(y,0), B_{in}-(1,0)\right) \Vert _{H_{x,y}^{2}}\le \epsilon \text {min}\{\nu ,\mu \}^\frac{1}{2}\) for some small \(\epsilon\) independent of \(\nu ,\mu\), then the solution of the system remains within \(\mathcal{O}(\text {min}\{\nu ,\mu \}^\frac{1}{2})\) of Couette flow, and close to Couette flow as \(t\rightarrow \infty\); the magnetic field remains within \(\mathcal{O}(\text {min}\{\nu ,\mu \}^\frac{1}{2})\) of the (1, 0), and close to (1, 0) as \(t\rightarrow \infty\).