December  2012, 5(6): 1091-1131. doi: 10.3934/dcdss.2012.5.1091

Stability and interaction of vortices in two-dimensional viscous flows

1. 

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

Received  December 2011 Revised  March 2012 Published  August 2012

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.
Citation: Thierry Gallay. Stability and interaction of vortices in two-dimensional viscous flows. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1091-1131. doi: 10.3934/dcdss.2012.5.1091
References:
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show all references

References:
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H. Abidi and R. Danchin, Optimal bounds for the inviscid limit of Navier-Stokes equations, Asymptot. Anal., 38 (2004), 35-46.  Google Scholar

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[20]

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[28]

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[29]

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[33]

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[37]

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[38]

T. Hmidi, Poches de tourbillon singulières dans un fluide faiblement visqueux, Rev. Mat. Iberoamericana, 22 (2006), 489-543. doi: 10.4171/RMI/464.  Google Scholar

[39]

J. Jiménez, H. K. Moffatt and C. Vasco, The structure of the vortices in freely decaying two-dimensional turbulence, J. Fluid Mech., 313 (1996), 209-222. doi: 10.1017/S0022112096002182.  Google Scholar

[40]

F. Jing, E. Kanso and P. Newton, Viscous evolution of point vortex equilibria: The collinear state, Phys. Fluids, 22 (2010), 123102. doi: 10.1063/1.3516637.  Google Scholar

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