Weak stability of the set of global solutions to the Navier-Stokes equations

Details Weak stability of the set of global solutions to the Navier-Stokes equations Weak stability of the set of global solutions to the Navier-Stokes equations

2011-09-19

arxiv.org/abs/1109.4043v1

Mathematics

Analysis of PDEs

Scientific paper

Let $\cG$ be the set of 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 $\cG$ belongs also to $\cG$ -- possibly after extracting a subsequence -- provided the convergence holds "anisotropically" in frequency space. Typically that 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 does not belong to $\cG$ (hence which produces a solution blowing up in finite time) cannot have a strong anisotropy in its frequency support.

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