Linear Stability, Inviscid Damping, and Scattering for Monotone Shear Flows in 2D Ideal MHD

We study the linearized 2D ideal incompressible magnetohydrodynamic equations around a monotone shear flow $(U(y),0)$ with a constant horizontal magnetic field $(β,0)$ on $\mathbb{T} \times \mathbb{R}$. For each fixed $β\neq 0$, we assume that the shear rate $U'$ is bounded above and below by positive constants and varies sufficiently slowly, although it need not be close to any constant. For every nontrivial perturbation with zero horizontal average, we establish sharp $L^2$ time rates in the corresponding paired norms: the horizontal velocity and magnetic components remain of order one, the vertical components decay at rate $\langle t \rangle^{-1}$, and the vorticity--current pair grows at rate $\langle t \rangle$. We prove strong $L^2$ scattering to two Alfvén profiles, identify the leading asymptotic behavior of the physical variables, and establish asymptotic completeness in the natural energy space.