Stability and interaction of vortices      in two-dimensional viscous flows

Thierry Gallay

doi:10.3934/dcdss.2012.5.1091

Article Contents

2012, Volume 5, Issue 6: 1091-1131. Doi: 10.3934/dcdss.2012.5.1091

This issue Previous Article Motivation, analysis and control of the variable density Navier-Stokes equations Next Article The thermo-mechanics of rate-type fluids

Stability and interaction of vortices in two-dimensional viscous flows

Thierry Gallay^1,

1.
Université de Grenoble 1, Institut Fourier, UMR 5582, B.P. 74，38402 Saint-Martin-d'Hères

Received: November 30, 2011

Revised: February 29, 2012

Published: November 2012

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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 $\mathbb{R}$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 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.

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Mathematics Subject Classification: Primary: 35Q30, 76D05; Secondary: 76D17, 76E07.

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