A NONLINEAR APPROXIMATION FOR VORTEX SHEET EVOLUTION AND SINGULARITY FORMATION

The evolution of a vortex sheet in two-dimensional, incompressible, inviscid flow is governed by the integro-differential equation of Birkhoff-Rott. We derive a simple approximation for vortex sheet evolution, consisting of a system of four first-order differential equations. This approximate system has the advantage of involving only local operators. The errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point. For the approximate equations, exact similarity solutions exhibiting singularity formation are constructed. © 1990.