The Hodograph Method in Fluid-Dynamics in the Light of Variational Inequalities

This paper concerns the problem of a flow past a given profile with prescribed velocity at infinity. In 1973 we announced (see [4]) a new approach-using variational inequalities-to the study in the hodograph plane of some twodimensional subsonic flows past a given convex profile. The aim of the present work is to give the proofs and develop further properties of the solution. For simplicity we confine ourselves here to the case of incompressible fluids. We plan to return to the general case in a forthcoming paper. For incompressible fluids, the existence and uniqueness of a flow past a given profile with prescribed velocity at infinity is well known. A complete bibliography can be found for example in the book of L. BERS [2]. For compressible fluids, the hodograph method has the notable advantage of giving linear equations, but it leadseven in the incompressible case to a free boundary value problem which looks at least as difficult as the original problem. Since no mathematical method was available, the difficulty was usually avoided by considering indirect problems, namely by determining a flow with a given hodograph. BERS [2] describes the situation by saying: "In general the boundary conditions become extremely complicated by going over to the hodograph plane, and this is even more pronounced if one uses the Legendre transform." Our main purpose here is to show that this difficulty can be overcome in some simple cases by using variational inequalities instead of equations. We would like to thank L. NIRENBERG for his crucial observations and useful suggestions while we were writing this paper. The plan of the work is the following: