Continuous subsonic-sonic potential flows in curved convergent nozzles

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle, which is a perturbation of a straight one, a given inlet, which is a perturbation of an arc centered at the vertex of the nozzle, and a given mass flux, 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. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only $ C^{1/2} $ Hölder continuous and the acceleration blows up.