Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics

We prove uniqueness and continuous dependence on initial data of weak solutions of the equations of compressible magnetohydrodynamics. The solutions we consider may exhibit discontinuities in density and in the gradients of velocity, temperature, and magnetic field. Continuous dependence is deduced by duality from existence and regularity of solutions of the adjoint of the first variation system. The analysis is complicated by the absence of strict parabolicity, the strong nonlinear coupling in the highest-order terms, and the lack of regularity in the coefficients of the adjoint system.