Ideal MHD. Part II: Rigidity from infinity for ideal Alfv\'en waves in 3D thin domains

This paper concerns the rigidity from infinity for Alfv\'en waves governed by ideal incompressible magnetohydrodynamic equations subjected to strong background magnetic fields along the $x_1$-axis in 3D thin domains $\Omega_\delta=\mathbb{R}^2\times(-\delta,\delta)$ with $\delta\in(0,1]$ and slip boundary conditions. We show that in any thin domain $\Omega_\delta$, Alfv\'en waves must vanish identically if their scattering fields vanish at infinities. As an application, the rigidity of Alfv\'en waves in $\Omega_{\delta}$, propagating along the horizontal direction, can be approximated by the rigidity of Alfv\'en waves in $\mathbb{R}^2$ when $\delta$ is sufficiently small. Our proof relies on the uniform (with respect to $\delta$) weighted energy estimates with a position parameter in weights to track the center of Alfv\'en waves. The key issues in the analysis include dealing with the nonlinear nature of Alfv\'en waves and the geometry of thin domains.