Global Smooth Sonic-Supersonic Flows in a Class of Critical Nozzles

Abstract

This paper concerns the global existence of smooth sonic-supersonic potential flows in a two-dimensional expanding nozzle with the critical geometry at the inlet. The flow is sonic and its velocity is along the normal direction at the inlet, which is a segment vertical to the walls of the nozzle, and is supersonic in the nozzle. Such a sonic-supersonic model is governed by a quasilinear nonstrictly hyperbolic equation with degeneracy at the inlet, and the degeneracy is strong in the sense that all characteristics from the inlet coincide with the inlet and never approach the domain. Furthermore, the asymptotic behaviors of the average speed on the cross section and the normal acceleration on the walls are of the same order with respect to the distance to the inlet, and of different order with respect to the height of the inlet. We seek the flows whose asymptotic behavior near the inlet is the same as the average speed on the cross section. An interesting phenomenon is that the existence of such sonic-supersonic flows depends on the height of the inlet. More precisely, it is shown by means of a method of characteristics with many elaborate estimates that there exists uniquely such a flow if it is suitably small, while it does not if it is suitably large. The results in the paper, together with other works, describe completely the geometry of the de Laval nozzles where there are smooth transonic flows of Meyer type whose sonic points are all exceptional.