Global stability and asymptotic behavior for incompressible ideal MHD equations with velocity damping term

TL;DR

This paper investigates the stability and long-term behavior of incompressible ideal MHD equations with velocity damping, revealing the stabilizing influence of the magnetic field and providing a general analytical framework for similar fluid models.

Contribution

It offers a new mathematical analysis of the stabilizing effects of magnetic fields in MHD with damping, extending understanding of asymptotic behavior in such systems.

Findings

Magnetic field stabilizes the system under small perturbations.

Established decay rates for solutions over time.

Developed a versatile analytical framework for partially dissipative fluids.

Abstract

In this article, we study the stability and large time behavior for an multi-dimensional incompressible magnetohydrodynamical system with a velocity damping term, for small perturbations near a steady-state of magnetic field fulfilling the Diophantine condition. Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time. Our proof approach mainly relies on the Fourier analysis and energy estimates. In addition, we provide a versatile analytical framework applicable to many other partially dissipative fluid models.