The Invariant Region For Non-isentropic Gas Dynamic Equations

We are mainly concerned about the invariant region for non-isentropic gas dynamic equations where p,u,p,e denote the density,velocity,pressure and internal energy respectively. We also as-sume that the gas is ideal and polytropic.Gas dynamic equations is one of the key issues in the research for conservation laws, in which the existence of global weak solutions with large initial data is one of the most important basic theoretic problems. In the study about the existence of global weak solutions for the equations of conservation laws, the theory of compensated compactness is the one of the most useful methods, for example, in the isentropic case, the weak solution existence for gas dynamic equations with cauchy data is completed using the compensated compactness theory, in the non-isentropic case, there hasn’t been virtual progress. Our main purpose in this paper is to study the basic problem of non-isentropic gas dynamic equations. We study the uniform boundedness of its weak solution by analyse the invariant region, we have proved that there doesn’t exist any bounded invariant region for non-isentropic gas equations, this conclusion indicates that there’s a virtual difficulty in the study about non-isentropic equations compared with the isentropic case.