Stability and Interaction of Vortices in Two-Dimensional Viscous Flows

TL;DR

This paper analyzes the stability and long-term behavior of vortices in two-dimensional viscous flows, proving existence of solutions and exploring the vanishing viscosity limit where vortex dynamics align with classical point vortex models.

Contribution

It provides a comprehensive mathematical analysis of vortex stability, existence of solutions for the Navier-Stokes equations with measure-valued vorticity, and the vortex behavior as viscosity approaches zero.

Findings

Existence of unique global solutions for all viscosities.

Vortex solutions converge to superpositions of Oseen vortices in the vanishing viscosity limit.

Stability analysis of Oseen vortices in high Reynolds number regimes.

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 flows. We consider the incompressible Navier-Stokes equations in the whole plane, 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 flow 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-Kirchhoff 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.