We study the compressible Euler equations with spherical symmetry surrounding a solid ball. For the spherically symmetric flow, the global existence of L ∞ entropy weak solutions has not yet obtained except a special case. In this paper, we prove the existence of global solutions in the more general case. We construct approximate solutions by using a modified Godunov scheme. The main point is to obtain an L ∞ bound for the approximate solutions.

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In this paper, we study the motion of isentropic gas in the Laval nozzle. The Laval nozzle is the most important type of nozzle utilized in some turbines. In particular, we consider unsteady flows, including transonic gas flows, and prove the existence of global solutions for the Cauchy problem. In spite of its importance, this problem has received little attention until now. The most difficult point is to obtain bounded estimates for approximate solutions. To overcome this, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation and yield a sharper bounded estimate. As a result, we find an invariant region for our solutions. Finally, in order to prove their convergence, we use the compensated compactness framework.

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