Continuous subsonic–sonic flows in a general nozzle

This paper concerns continuous subsonic–sonic potential flows in a two dimensional finite nozzle with a general upper wall and a straight lower wall. We give a class of nozzles where continuous subsonic–sonic flows may exist. Consider a continuous subsonic–sonic flow in such a nozzle after rescaling the upper wall in a small scale. It is shown that for a given inlet and a fixed point at the upper wall, there exists uniquely a continuous subsonic–sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the fixed point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C1/2C1/2 Hölder continuous and the acceleration blows up at the sonic state. As the scale tends to zero, the precise convergent rate of the continuous subsonic–sonic flow converging to the sonic state is also determined.