Stability and interaction of vortices in two-dimensional viscous flows

Abstract

The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous ows. We consider the incompressible Navier-Stokes equations in the whole plane ℝ 2, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial ow is a finite collection of point vortices. In that situation, we show that the solution behaves, in the vanishing viscosity limit, as a superposition of Oseen vortices whose centers evolve according to the Helmholtz-Kirchho point vortex system. The proof requires a careful stability analysis of the Oseen vortices in the large Reynolds number regime, as well as a precise computation of the deformations of the vortex cores due to mutual interactions.