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

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.