On the stability in weak topology of the set of global solutions to the Navier-Stokes equations

Hajer Bahouri, Isabelle Gallagher

Published 2011-09-19, updated 2013-02-22Version 2

Let $X$ be a suitable function space and let $\cG \subset X$ 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 $\cG$ belongs to $\cG$ if $n$ is large enough, 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.