On the Decay and Stability of Global Solutions to the 3D Inhomogeneous MHD system

TL;DR

This paper studies the long-term decay and stability of smooth solutions to the 3D inhomogeneous MHD system, providing explicit decay rates and demonstrating stability under small perturbations.

Contribution

It establishes decay rates for solutions and their derivatives, and proves stability of large solutions under small initial perturbations, highlighting differences from Navier-Stokes equations.

Findings

Velocity and magnetic fields decay to zero with explicit rates.

Small perturbations of large solutions remain close and produce global smooth solutions.

Derived new elliptic estimates for coupled velocity and magnetic fields.

Abstract

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a, u, B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a, u, B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\|(u, B)\|_{L^{1}(\mathbb{R}^{+};\dot{B}_{2,1}^{5/2})} < C$, which is different to Navier-Stokes equations.