Stability Threshold for 2D Shear Flows of Generalized Magnetohydrodynamic Equations Near Couette

In this paper, we investigate the nonlinear stability threshold for the shear flows of the generalized magnetohydrodynamic (GMHD) equations on $${\mathbb {T}} \times {\mathbb {R}}$$ . We prove the nonlinear stability of the shear flow $$\left( U_s, B_s\right) =\left( (e^{-t\nu \partial _{y}^4}U(y),0)^{\top },(\alpha ,0)^{\top }\right)$$ with the initial data of the shear flow (U(y), 0) close to the Couette flow (y, 0). For sufficiently large $$|\alpha |$$ , we prove that when the initial perturbations satisfy $$\Vert (u_{in},b_{in})\Vert _{H^{N+1}}=\epsilon \ll \nu ^{\frac{7}{10}+{\tilde{\delta }}}$$ for any fixed $${\bar{\delta }}>0$$ , here $$\nu$$ is a positive real parameter, then for all $$t> 0$$ , the global solution of the 2D GMHD equations remains $$\nu ^{-\frac{1}{5}-\frac{{\bar{\delta }}}{2}}$$ $$\epsilon$$ -close to $$U_s=(e^{-t\nu \partial _{y}^4}U(y),0)^{\top }.$$