Large Solutions to the Initial-Boundary Value Problem for Planar Magnetohydrodynamics

Abstract

An initial-boundary value problem for nonlinear magnetohydrodynamics (MHD) in one space dimension with general large initial data is investigated. The equations of state have nonlinear dependence on temperature as well as on density. For technical reasons the viscosity coefficients and magnetic diffusivity are assumed to depend only on density. The heat conductivity is a function of both density and temperature, with a certain growth rate on temperature. The existence, uniqueness, and regularity of global solutions are established with large initial data in H1 . It is shown that no shock wave, vacuum, or mass or heat concentration will be developed in a finite time, although the motion of the flow has large oscillations and there is a complex interaction between the hydrodynamic and magnetodynamic effects.