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Three-Dimensional Instability of Planar Flows

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Abstract

We study the stability of two-dimensional solutions of the three-dimensional Navier–Stokes equations, in the limit of small viscosity. We are interested in steady flows with locally closed streamlines. We consider the so-called elliptic and centrifugal instabilities, which correspond to the continuous spectrum of the underlying linearized Euler operator. Through the justification of highly oscillating Wentzel–Kramers–Brillouin expansions, we prove the nonlinear instability of such flows. The main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions.

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Authors and Affiliations

  1. CNRS, Laboratoire J.-A Dieudonne, UMR 6621, Université de Nice, Parc Valrose, 06108, Nice Cedex 02, France

    F. Gallaire

  2. CNRS, DMA, UMR 8553, Ecole Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France

    D. Gérard-Varet

  3. CNRS, Laboratoire J.-A Dieudonne, UMR 6621, Université de Nice, Parc Valrose, 06108, Nice Cedex 02, France

    F. Rousset

Authors
  1. F. Gallaire
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  2. D. Gérard-Varet
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  3. F. Rousset
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Correspondence to F. Gallaire.

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Communicated by C. M. Dafermos

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Gallaire, F., Gérard-Varet, D. & Rousset, F. Three-Dimensional Instability of Planar Flows. Arch Rational Mech Anal 186, 423–475 (2007). https://doi.org/10.1007/s00205-007-0072-6

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  • DOI: https://doi.org/10.1007/s00205-007-0072-6

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