Global existence and stability of viscous Alfv\'en waves in the large-box limit for MHD systems

Abstract

This paper rigorously analyzes how the {\it large box limit} fundamentally alters the global existence theory and dynamics behavior of the incompressible magnetohydrodynamics (MHD) system with small viscosity/resistivity $(0<\mu\ll 1)$ on periodic domains $Q_L=[-L,L]^3$, in presence of a strong background magnetic field. While the existence of global solutions (viscous Alfv\'en waves) on the whole space $\R^3$ was previously established in \cite{He-Xu-Yu}, such results cannot be expected for general finite periodic domains. We demonstrate that global solutions do exist on the torus $Q_L=[-L,L]^3$ precisely when the domain exceeds a size $L_\mu>e^{\f1\mu}$, providing the first quantitative characterization of the transition to infinite-domain-like behavior.