Shock waves in the nonisentropic gas flow

In this paper we propose an extended entropy condition for general systems of hyperbolic conservation laws with several space variables. This entropy condition generalizes the well-known condition (E) of Volpert for a single conservation law with several space variables and reduces to the entropy condition proposed earlier by the author for systems with one space variable. The Riemann problem for general nonisentropic gas equations has a unique solution for initial data with arbitrarily large jumps. The occurrence of a vacuum region is observed. The projections of shock curves on the pressure-velocity plane are analyzed so as to study the interaction of weak shocks. Our results differ markedly from those of previous works in that we do not assume the equation of state to be polytropic. In fact our assumptions on the equation of state allow the pressure to be a nonconvex function of specific volume.The Riemann problem for this general system of gas equations was also treated by B. Wendroff when the initial data are near constant.