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

TL;DR

This paper investigates how increasing the size of a periodic domain affects the existence and stability of viscous Alfvén waves in incompressible MHD systems with small viscosity, revealing a critical domain size for global solutions.

Contribution

It provides the first quantitative characterization of the domain size threshold beyond which global solutions resemble those on the whole space, highlighting the impact of the large-box limit.

Findings

Global solutions exist when domain size exceeds $L_>e^{1/}$.

The large-box limit induces a transition in the existence theory.

Results connect finite periodic domains with infinite space behavior.

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.