Stability of vortex lines attached to airfoils in subsonic Euler flows

We study a long-standing problem about two-dimensional subsonic flows past airfoils attached by vortex lines. The flows are governed by steady compressible Euler equations. The vortex lines attached to the two edge points of airfoils are treated as free boundaries. The location of the vortex lines and the asymptotic direction of the flows are determined through solving a functional equation by employing the implicit function theorem. Various barrier functions are constructed to obtain estimates in weighted Hölder norms so that both the regularity of the flows near the edge points and the asymptotic behavior at infinity are controlled. This paper gives the first result about the airfoil problems containing vortex lines in a whole plane.