Green’s function for a general system of hyperbolic-parabolic balance laws with applications

We consider a general system of hyperbolic-parabolic balance laws, which contains both second-order dissipation represented by a viscosity matrix and lower-order damping or relaxation. As dictated by physics, the viscosity matrix and the Jacobian matrix of the lower-order term are usually rank deficient. We study the Green’s function of its linearization around a constant equilibrium state. The Green’s function is constructed using heat kernels along equilibrium characteristic directions and delta functions along other predetermined directions. Detailed error estimates are given in the space-time pointwise sense. Our main result is then applied to two different models of polyatomic gas flows. The first one is a model of translational and vibrational non-equilibrium flows with a general setting in the fundamental thermodynamic relations. The second one is a two-temperature Navier–Stokes type model for polyatomic gas flows, proposed by Aoki et al., with specific equations of state. For each model, we give an explicit formulation of Green’s function for the Cauchy problem.