Systems of partial differential equations of mixed hyperbolic–parabolic type

We prove the global existence and uniqueness of solutions of certain mixed hyperbolic–parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier–Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new.