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

Abstract

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 $$\bigcup _{t\in (T-1,T)}(B(0,R)\cap {\mathbb{R}}_{+}^3)\times \{t\},R=O(\sqrt{T-t})$$ 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.

arXiv comment38 pages, 1 figure

References in corpus (8)

• Global weak Besov solutions of the Navier-Stokes equations and applications
• Estimates for the Navier-Stokes equations in the half-space for non localized data
• Ancient Solutions to Navier-Stokes Equations in Half Space
• On local Type I singularities of the Navier-Stokes equations and Liouville theorems
• Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities
• Local energy weak solutions for the Navier-Stokes equations in the half-space
• On Vortex Alignment and Boundedness of $L^q$ Norm of Vorticity
• On Type I Blowups of Suitable Weak Solutions to Navier-Stokes Equations near Boundary