Global existence of strong solutions to MHD with density-depending viscosity and temperature-depending heat-conductivity in unbounded domains

We study the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature. In particular, we obtain the global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. This result generalizes the classical one of the compressible Navier–Stokes system with constant viscosity and heat conductivity [Kazhikhov, Siberian. Math. J. 23, 44–49 (1982)] to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic and hydrodynamic effects is complex and the motion of the flow has large oscillations.