Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions

Autor: Christophe Prange, Tobias Barker
Přispěvatelé: Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: Barker, T & Prange, C 2019, ' Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions ', Archive for Rational Mechanics and Analysis, vol. 235, pp. 881–926 . https://doi.org/10.1007/s00205-019-01435-z
Archive for Rational Mechanics and Analysis
Archive for Rational Mechanics and Analysis, Springer Verlag, 2019, ⟨10.1007/s00205-019-01435-z⟩
ISSN: 0003-9527
1432-0673
DOI: 10.1007/s00205-019-01435-z
Popis: 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.
Comment: 38 pages, 1 figure
Databáze: OpenAIRE
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