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Interaction of Vortices in Weakly Viscous Planar Flows

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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 ν, 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 ν → 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 us 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 makes it possible to estimate the self-interactions, which play an important role in the convergence proof.

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  1. Institut Fourier, Université de Grenoble I, B.P. 74, 38402, Saint-Martin-d’Hères, France

    Thierry Gallay

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Communicated by A. Mielke

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Gallay, T. Interaction of Vortices in Weakly Viscous Planar Flows. Arch Rational Mech Anal 200, 445–490 (2011). https://doi.org/10.1007/s00205-010-0362-2

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