Asymptotic stability of Landau solutions to the MHD system and energy decay

TL;DR

This paper proves the asymptotic stability of Landau solutions in the 3D incompressible MHD system and derives explicit decay rates for perturbations under certain conditions.

Contribution

It establishes the $L^2$-stability of Landau solutions and provides explicit algebraic decay rates for velocity and magnetic field perturbations.

Findings

Weak solutions are $L^2$-asymptotically stable around Landau solutions.

Under additional conditions, explicit algebraic decay rates are obtained.

Stability holds for solutions satisfying a strong energy inequality.

Abstract

We consider the three-dimensional incompressible MHD system. Any weak solution satisfying a strong energy inequality is $L^2$-asymptotically stable around a Landau solution. Under an additional integrability assumption on the initial perturbation, we also obtain an explicit algebraic decay rate for the $L^2$-norm of the velocity and magnetic perturbations.