On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfv\'en waves

TL;DR

This paper analyzes the global behavior of solutions to 3D incompressible magnetohydrodynamics, demonstrating long-time persistence of wave-like behavior and eventual damping due to diffusive effects without symmetry assumptions.

Contribution

It provides a rigorous mathematical framework for the nonlinear stability of Alfvén waves in 3D MHD, with a novel energy and geometric analysis approach.

Findings

Solutions behave like non-dispersive waves initially

Solutions are damped and decay fast after long-time accumulation of diffusion

The analysis does not depend on symmetry or vorticity conditions

Abstract

We construct and study global solutions for the 3-dimensional incompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution initially behaves like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number), thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects, eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any condition on the symmetry or on the vorticity. The size of data and the a priori estimates do not depend on viscosity. The proof is builded upon a novel use of the basic energy identity and a geometric study of the characteristic hypersurfaces. The approach is partly inspired by Christodoulou-Klainerman's proof of the nonlinear stability of Minkowski space in general relativity.