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On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

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Abstract

Let X be a suitable function space and let \({\mathcal{G} \subset X}\) be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of \({\mathcal{G}}\) belongs to \({\mathcal{G}}\) if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to \({\mathcal{G}}\) (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.

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Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris 12, 61, avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Hajer Bahouri

  2. Institut de Mathématiques de Jussieu - Paris Rive Gauche UMR CNRS 7586, Université Paris-Diderot (Paris 7), Bâtiment Sophie Germain Case 7012, 75205, Paris Cedex 13, France

    Isabelle Gallagher

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  1. Hajer Bahouri
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  2. Isabelle Gallagher
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Correspondence to Isabelle Gallagher.

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Communicated by V. Šverák

Isabelle Gallagher was partially supported by the A.N.R grant ANR-08-BLAN-0301-01 “Mathocéan”, and the Institut Universitaire de France.

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Bahouri, H., Gallagher, I. On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations. Arch Rational Mech Anal 209, 569–629 (2013). https://doi.org/10.1007/s00205-013-0623-y

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