Invariant region on a non-isentropic gas dynamics system

In this paper, we found a special non-isentropic gas dynamics system, whose invariant region is the opposite of the corresponding isentropic case. This shows that the powerful invariant region theory introduced by Chueh, Conley and Smoller for general hyperbolic system of two conservation laws cannot be obviously applied to obtain the a priori L∞ estimates for systems of more than two equations.