Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity

Abstract

In this study, we consider the initial boundary value problem of the planar magnetohydrodynamics (MHD) system when the viscous coefficients and heat conductivity depend on the temperature, which are assumed to be proportional to θα{\theta }^{\alpha }, α≥0\alpha \ge 0, and θβ{\theta }^{\beta }, β≥0\beta \ge 0, respectively, and magnetic diffusivity coefficient depends on the specific volume. We prove the existence and uniqueness of the global-in-time classical solution with general large initial data provided that α\alpha is sufficiently small and there is no restriction on the parameter β\beta . Moreover, the nonlinearly exponential stability of the solution is obtained. As a result, we extend the works given by Sun et al. for the full compressible Navier-Stokes equations and by Li and Shang for the full compressible MHD equations.

Keywords

• mhd equations
• global classical solution
• temperature-dependent coefficients
• exponential stability
• 35q35
• 76n10
• 76w05