Global existence theory for general hyperbolic-parabolic balance laws with application

Abstract

We study a general system of hyperbolic-parabolic balance laws in [Formula: see text] space dimensions ([Formula: see text]). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.