Smooth Transonic Flows of Meyer Type in De Laval Nozzles

A smooth transonic flow problem is formulated as follows: for a de Laval nozzle, one looks for a smooth transonic flow of Meyer type whose sonic points are all exceptional and whose flow angle at the inlet is prescribed. If such a flow exists, its sonic curve must be located at the throat of the nozzle and the nozzle should be suitably flat at its throat. The flow is governed by a quasilinear elliptic-hyperbolic mixed type equation and it is strongly degenerate at the sonic curve in the sense that all characteristics from the sonic points coincide with the sonic curve and never approach the supersonic region. For a suitably flat de Laval nozzle, the existence of a local subsonic-sonic flow in the convergent part and a local sonic-supersonic flow in the divergent part is proved by some elaborate elliptic and hyperbolic estimates. The precise asymptotic behavior of these two flows near the sonic state is shown and they can be connected to a smooth transonic flow whose acceleration is Lipschitz continuous. The flow is also shown to be unique by an elaborate energy estimate. Moreover, we give a set of infinitely long de Laval nozzles, such that each nozzle admits uniquely a global smooth transonic flow of Meyer type whose sonic points are all exceptional, while the same result does not hold for smooth transonic flows of Meyer type with nonexceptional points.