Global existence and stability to the polytropic gas dynamics with an outer force

Abstract

In this paper, we apply the viscosity-flux approximation method coupled with the maximum principle to obtain the a-priori L∞ estimates for the approximation solutions of the compressible polytropic gas dynamics system with an outer force, and prove the global existence of entropy solutions for any adiabatic exponent γ>1.The present problem was classical and the global existence was known long ago. However, the L∞ estimates grow larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. We thus prove the uniformly L∞ estimate with respect to the time variable. This means that our solutions are stable. The key idea is to employ an invariant region depending on the space and time variables. This method enables us to investigate the behavior of solutions in detail and deduce the desired estimates.