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 a 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 $$\begin{aligned} \bigcup _{t\in (T-1,T)} \big (B(0,R)\cap {\mathbb {R}}^3_{+}\big )\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t}), \end{aligned}$$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 et al. [20]. 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.