The invariant region for the special gas dynamics system

In this paper, we study the invariant region for the special gas dynamics system, where the conservation law of energy in classical Euler equations is replaced by the specific entropy. Firstly, the Riemann solutions for five domains in the u-p plane are completely constructed. Then we obtain the bounded convex invariant region of the Riemann solutions through analyzing structure of the Riemann solutions. Moreover, with the bounded convex invariant region, we get the uniform boundedness of approximate solutions, which is the foundation to prove the existence of weak solutions for the special gas dynamic system using compensated compactness method.