Large-time Behavior of Magnetohydrodynamics with Temperature-Dependent Heat-Conductivity

TL;DR

This paper analyzes the long-term behavior of solutions to a magnetohydrodynamic system with temperature-dependent heat conductivity, proving boundedness and exponential stability over time.

Contribution

It extends previous results by establishing boundedness and stability for MHD equations with nonlinear, temperature-dependent heat conductivity.

Findings

Boundedness of specific volume and temperature over time

Exponential stability of the global strong solution

Generalization of previous Navier-Stokes results to MHD systems

Abstract

We study the large-time behavior of strong solutions to the equations of a planar magnetohydrodynamic compressible flow with the heat conductivity proportional to a nonnegative power of the temperature. Both the specific volume and the temperature are proved to be bounded from below and above independently of time. Moreover, it is shown that the global strong solution is nonlinearly exponentially stable as time tends to infinity. Our result can be regarded as a natural generalization of the previous ones for the compressible Navier-Stokes system to MHD system with either constant heat-conductivity or nonlinear and temperature-dependent heat-conductivity.