On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations
Identifieur interne : 000021 ( Main/Exploration ); précédent : 000020; suivant : 000022On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations
Auteurs : Hajer Bahouri [France] ; Isabelle Gallagher [France]Source :
- Archive for Rational Mechanics and Analysis [ 0003-9527 ] ; 2013-08-01.
Abstract
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.
Url:
DOI: 10.1007/s00205-013-0623-y
Affiliations:
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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.</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Île-de-France</li></region><settlement><li>Créteil</li><li>Paris</li></settlement></list><tree><country name="France"><region name="Île-de-France"><name sortKey="Bahouri, Hajer" sort="Bahouri, Hajer" uniqKey="Bahouri H" first="Hajer" last="Bahouri">Hajer Bahouri</name></region><name sortKey="Bahouri, Hajer" sort="Bahouri, Hajer" uniqKey="Bahouri H" first="Hajer" last="Bahouri">Hajer Bahouri</name><name sortKey="Gallagher, Isabelle" sort="Gallagher, Isabelle" uniqKey="Gallagher I" first="Isabelle" last="Gallagher">Isabelle Gallagher</name><name sortKey="Gallagher, Isabelle" sort="Gallagher, Isabelle" uniqKey="Gallagher I" first="Isabelle" last="Gallagher">Isabelle Gallagher</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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