Special Session 20: Partial Differential Equations from Fluid Mechanics and Mathematical Physics

In past work, the author has developed the Operator Method to study the existence of convex classical solutions to convex one-layer and multi-layer freeboundary problems in fluid dynamics (see Trans. Amer. Math. Soc. 350(1998), pp. 2981-3020, for example). The general idea was to obtain the convex free boundary either as a convex functional minimizer (convex variational inequalities), or as the limit of a sequence of convex solutions of a corresponding convex fixed point problem for a one-parameter family of convexity-preserving free-boundary-perturbation operators. The purpose of the present paper is to show that the same operator method, based on the same operators, can be generalized to study geometricallygeneral double-free-boundary problems for annular flows of ideal fluid in two space dimensions, in which the free-boundaries are characterized by a generalized form of Bernoulli’s law. This is the first time that the Operator Method has been adapted to an essentially arbitrary geometric context.