The well-posedness of axially symmetric compressible subsonic cavity flow

In this paper, we investigate the existence and uniqueness of the three-dimensional axially symmetric steady compressible subsonic flow behind an obstacle. It is shown that there exists a critical value M-cr > 0 such that if the total incoming mass flux M-0 is less than M-cr, then there exists a unique subsonic cavity flow, with the free boundary smoothly emanating from the endpoint of the obstacle and having a constant end pressure along the free boundary. Moreover, the positivity of the radial velocity and the asymptotic behavior of the cavity flow at far field are also obtained. As a by-product, the convexity of the free boundary is derived. Based on the convex geometry property of the free boundary, we obtain the positivity of the axial velocity in the fluid. If the free boundary is strictly convex, it is further proved that the optimal regularity of the free boundary at the separation point is C-1,C-1/2.