$L^{\mathrm{\infty }}$ solutions for a model of polytropic gas flow with diffusive entropy

We establish the global existence of $L^{\mathrm{\infty }}$ solutions for a model of polytropic gas flow with diffusive entropy. The result is obtained by showing the convergence of a class of finite difference schemes, which includes the Lax-Friedrichs and Godunov schemes. Such convergence is achieved by proving the estimates required for the application of the compensated compactness theory.