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Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

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Abstract

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called “Oseen’s vortex”. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.

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Authors and Affiliations

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

    Thierry Gallay

  2. Department of Mathematics and Center for BioDynamics, Boston University, 111 Cummington St., Boston, MA, 02215, USA

    C. Eugene Wayne

Authors
  1. Thierry Gallay
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  2. C. Eugene Wayne
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Communicated by A. Kupiainen

Acknowledgement The first author is indebted to J. Dolbeault and, especially, to C. Villani for suggesting the beautiful idea of using the Boltzmann entropy functional in the context of the two-dimensional Navier-Stokes equation. The research of C.E.W. is supported in part by the NSF under grant DMS-0103915.

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Gallay, T., Wayne, C. Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation. Commun. Math. Phys. 255, 97–129 (2005). https://doi.org/10.1007/s00220-004-1254-9

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