On Global-in-time Solutions of Incompressible MHD Equations with Small Alfv\'en Numbers

TL;DR

This paper proves the existence of global-in-time solutions for incompressible MHD equations with small Alfvén numbers, extending previous results to the case where both viscosity and resistivity are positive, using a new analytical approach.

Contribution

It introduces a novel method with a key bilinear estimate to establish global solutions for MHD with positive viscosity and resistivity, generalizing prior work.

Findings

Established global-in-time solutions for small Alfvén numbers with positive viscosity and resistivity.

Developed a new analytical approach including a bilinear estimate for nonlinear terms.

Analyzed the vanishing nonlinear interaction and small Alfvén number limits of solutions.

Abstract

In 1965 Kraichnan pointed out that a sufficiently strong background magnetic field, i.e. the case of small Alfv\'en number, will reduce the nonlinear interaction and inhibit the formation of strong gradients in the magnetohydrodynamic (abbr. MHD) system with $\mu=\nu\geqslant 0$, where ${\mu}$ and $\nu $ are the coefficients of kinematic viscosity and resistivity resp.. This means that the MHD system with ${\mu}=\nu\geqslant 0$ admits global-in-time large perturbation solutions with small Alfv\'en numbers. The existence of such large perturbation solutions was first mathematically verified in H\"older spaces by Bardos--Sulem--Sulem for the case ${\mu}=\nu= 0$ in 1988, and in Sobolev spaces by Cai--Cui--Jiang--Liu for the case ${\mu}=\nu> 0$ recently. In this paper, we further found a similar result for the general case ``${\mu}>0$ and $\nu>0$", and provide a rigorous proof by developing a new approach, which includes a key bilinear estimate for dealing with the nonlinear interaction terms. Moreover both additional results for the vanishing behavior of the nonlinear interaction and the small Alfv\'en number limit of solutions are also established.