BARKER, Tobias; PRANGE, Christophe

doi:10.1007/s00205-019-01435-z

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BARKER, Tobias

PRANGE, Christophe
Institut de Mathématiques de Bordeaux [IMB]
Centre National de la Recherche Scientifique [CNRS]

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Article de revue

Ce document a été publié dans

Archive for Rational Mechanics and Analysis. 2019-06-19, vol. 235, p. 881–926

Springer Verlag

Résumé en anglais

This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in $\mathbb{R}^3_{+}\times (0,T)$ with no-slip boundary condition, with the assumption that the solution satisfies the `ODE blow-up rate' Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets of $$\bigcup_{t\in(T-1,T)} \big(B(0,R)\cap\mathbb{R}^3_{+}\big)\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t})$$ where the vorticity has large magnitude, then $(0,T)$ is a regular point. This result is inspired by and improves the regularity criteria given by Giga, Hsu and Maekawa (2014). We also obtain new local versions for suitable weak solutions near the flat boundary. Our method hinges on new scaled Morrey estimates, blow-up and compactness arguments and `persistence of singularites' on the flat boundary. The scaled Morrey estimates seem to be of independent interest.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190359

DOI

http://dx.doi.org/10.1007/s00205-019-01435-z

Project ANR

Bords, oscillations et couches limites dans les systèmes différentiels - ANR-16-CE40-0027
Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure - ANR-18-CE40-0027

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251