2011/09/19 by Hajer Bahouri, Isabelle Gallagher, Bahouri, Hajer +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1109.4043
openalex publication_date 2011/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a suitable function space and let \cG ⊂ 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.