Desingularizations of systems of point vortices

Abstract

Consider the evolution of an incompressible inviscid fluid in R 2. Zabusky conjectured that: given a steady system of point vortices there exists a one-parameter family of steady system of vortex patches in which each patch shrinks to the corresponding point vortex while keeping the same circulation as the parameter tends to zero. Such a family is called a desingularization of the given system of point vortices. In this article, we establish that: 1. (a) Zabusky's conjecture holds under a mild condition on the system of point vortices; 2. (b) the desingularizations of the system of point vortices preserve their type of stability; 3. (c) desingularizations of an unfolding of a system of point vortices exist under a “typical” situation.