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