Communications in Mathematical Sciences

Volume 22 (2024)

Number 5

Global existence and exponential stability of planar magnetohydrodynamics with temperature-dependent transport coefficients

Pages: 1251 – 1285

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n5.a4

Authors

Ying Sun (Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical)

Jianwen Zhang (School of Mathematical Sciences, Xiamen University, Xiamen, China)

Abstract

This paper is concerned with an initial and boundary value problem for planar compressible magnetohydrodynamics with temperature-dependent transport coefficients. In the case when the viscosity $\mu (\theta) = \lambda (\theta) = \theta^\alpha$, the magnetic diffusivity $\nu (\theta ) = \theta^\alpha$ and the heat-conductivity $\kappa (\theta ) = \theta^\beta$ with $\alpha ,\beta \in[0, \infty)$, we prove the global existence of strong solution under some restrictions on the growth exponent $\alpha$ and the initial norms. As a byproduct, the exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if $\alpha \geq 0$ is small, and the growth exponent of heat-conductivity $\beta \geq 0$ can be arbitrarily large.

Keywords

planar magnetohydrodynamics, global strong solutions, temperature-dependent transport coefficients, exponential stability, large data

2010 Mathematics Subject Classification

35Q35, 76N10

Received 20 May 2023

Accepted 6 November 2023

Published 15 July 2024