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

2013; Springer Science+Business Media; Volume: 209; Issue: 2 Linguagem: Inglês

10.1007/s00205-013-0623-y

1432-0673

Hajer Bahouri, Isabelle Gallagher,

Advanced Mathematical Physics Problems

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.