On the Equation Satisfied by a Steady Prandtl-Munk Vortex Sheet

Abstract

We show the the voricity distribution obtained by minimizing the induced drag on a wing, the so called Prandtl-Munk vortex sheet, is not a travelling-wave weak solution of the Euler equations, contrary to what has been claimed by a number of authors. Instead, it is a weak solution of a non-homogeneous Euler equation, where the forcing term represents a "tension" force applied to the tips. This is consistent with a heuristic arguement due to Saffman. Thus, the notion of weak solution captures the correct physical behavior in this case.