Mathema Tics: M. Shiffman on the Existence of Subsonic Flows of a Compressible Fl Uid

1. The stationary, irrotational flow of a compressible fluid is characterized by a quasi-linear second order partial differential equation with coefficients which are functions of the first derivatives. The equation is sometimes elliptic and sometimes hyperbolic, depending on the magnitude of these first derivatives. The question of the existence of such flows satisfying prescribed conditions is uncertain, and it is not known beforehand whether, for the desired flow, the equation is always elliptic or partly elliptic and partly hyperbolic. We shall indicate here how to establish existence theorems for purely subsonic flows, corresponding to the completely elliptic case. Consider, for definiteness, the two-dimensional flow about a fixed object with a prescribed velocity u0 at infinity in the positive x-direction. The boundary of the fixed object will be supposed to be a smooth curve, say with continuous third derivatives. The fluid will be supposed to have the adiabatic law p = Apy, y > 1, as equation of state, although the following considerations apply to any equation of state. The flow is characterized by a velocity potential function sp(x, y) satisfying the partial differential equation