Interaction of vortices in viscous planar flows

TL;DR

This paper studies the inviscid limit of 2D incompressible Navier-Stokes equations with initial point vortices, showing convergence to a superposition of Lamb-Oseen vortices with centers evolving via a viscous regularization of the vortex system.

Contribution

It proves the convergence of Navier-Stokes solutions to a superposition of Lamb-Oseen vortices in the inviscid limit, with detailed vortex deformation analysis.

Findings

Convergence of Navier-Stokes to Lamb-Oseen vortices as viscosity approaches zero

Explicit description of vortex deformation due to mutual interactions

Estimation of self-interactions critical for convergence proof

Abstract

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.