Existence of global solutions for isentropic gas flow in a divergent nozzle with friction

We study isentropic gas flow in a divergent nozzle with friction. It is physically known that the flow is influenced by friction. In this paper, we investigate the mathematical effect of the friction on solutions. We first construct approximate solutions by using the Godunov scheme and the fractional step procedure. We obtain the bounded estimate by the invariant region. This is the difficult point of the present paper. We use the compensated compactness to prove their convergence. Finally, we prove the existence of global solutions with arbitrary L∞L∞ data.