Global L∞ entropy solutions to system of polytropic gas dynamics with a source

Abstract

In this paper, we study the global $L^{\infty }$ entropy solutions for the Cauchy problem of the polytropic gas dynamics system in a general nozzle with friction. First, under bounded conditions on the $L^1$ norm of the cross-sectional area function $A\left(x\right)$ and the friction function $\alpha \left(x\right)$, we apply the flux-approximation technique coupled with the classical viscosity method to obtain the $L^{\infty }$ estimates of the viscosity-flux approximate solutions for any exponent $\gamma \ge 1$; Second, by using the compactness framework from the compensated compactness theory, we prove the convergence of the viscosity-flux approximate solutions and obtain the global existence of the $L^{\infty }$ entropy solutions.