Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics

Abstract

The global (in time) existence and asymptotic stability of smooth solutions to the initial value problem are proved for a general class of quasilinear symmretric hyperbolic-parabolic composite, systems, under the smallness assumptions on the initial data and the dissipation condition on the linearized systems. In the special case of hyperbolic-parabolic systems of conservation laws with a convex entropy, it is also proved that for time t → ∞, the solutions of the nonlinear systems are asymptotic to those of the linear ones if the space-dimension n ≥ 2, and to those of the semi-linear ones if n = 1. These results are applicable to the equations of compressible viscous fluids, the equations of magnetohydrodynamics (or electro-magneto-fluid dynamics) for electrically conducting compressible viscous fluids, the equations of heat conduction with finite speed of propagation, and so on. Furthermore hyperbolic systems of conservation laws with small viscosity are investigated on the relation to the limit systems without viscosity. It is proved that as viscosity tends to zero, the smooth solutions of the systems with viscosity converge on a finite time interval to the smooth solutions of the limit systems.