Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity

Neustupa, Jiří; Penel, Patrick

doi:10.1016/j.crma.2012.06.008

Partial Differential Equations/Mathematical Problems in Mechanics

Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity
[Critères de régularité des solutions faibles des équations de Navier–Stokes fondés sur les projections spectrales de la vorticité]

Présenté par : Gérard Iooss

Neustupa, Jiří ¹ ; Penel, Patrick ²

¹ Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
² Université du Sud, Toulon-Var, Mathématique, BP 20132, 83957 La Garde cedex, France

Comptes Rendus. Mathématique, Tome 350 (2012) no. 11-12, pp. 597-602.

Résumé
Abstract

Soit ${E_{λ}}$ la résolution spectrale de lʼidentité associée à lʼopérateur auto-adjoint curl dans lʼespace $L_{σ}^{2} (R^{3})$ , et soit $a = a (t)$ une fonction à valeurs réelles définie sur $(0, T)$ . On note $P_{a}^{+} : = \int_{a}^{\infty} d E_{λ}$ , puis, lorsque v est solution faible du problème de condition initiale de Navier–Stokes dans $R^{3} \times (0, T)$ , $ω_{a}^{+} : = P_{a}^{+} curl v$ . On établit alors la régularité de v sous certaines conditions imposées à a et, ou bien à $ω_{a}^{+}$ , ou bien à sa troisième composante $ω_{a 3}^{+}$ seulement.

We denote $P_{a}^{+} : = \int_{a}^{\infty} d E_{λ}$ , where ${E_{λ}}$ is the spectral resolution of identity associated with the self-adjoint operator curl in the space $L_{σ}^{2} (R^{3})$ . Further, we denote $ω_{a}^{+} : = P_{a}^{+} curl v$ , where v is a weak solution to the Navier–Stokes initial value problem in $R^{3} \times (0, T)$ . We assume that $a = a (t)$ is a real function in $(0, T)$ . We show that certain conditions imposed on function a and $ω_{a}^{+}$ , or only on the third component $ω_{a 3}^{+}$ of $ω_{a}^{+}$ , imply regularity of solution v.

Reçu le : 2012-03-27
Accepté le : 2012-06-21
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.06.008

Affiliations des auteurs :

Neustupa, Jiří ¹ ; Penel, Patrick ²

¹ Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
² Université du Sud, Toulon-Var, Mathématique, BP 20132, 83957 La Garde cedex, France

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     title = {Regularity criteria for weak solutions to the {Navier{\textendash}Stokes} equations based on spectral projections of vorticity},
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Neustupa, Jiří; Penel, Patrick. Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity. Comptes Rendus. Mathématique, Tome 350 (2012) no. 11-12, pp. 597-602. doi : 10.1016/j.crma.2012.06.008. http://www.numdam.org/articles/10.1016/j.crma.2012.06.008/

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