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This paper is concerned with geometric regularity criteria for theNavier-Stokes equations in $\mathbb{R}^3_{+}\times (0,T)$ with no-slip boundarycondition, with the assumption that the solution satisfies the `ODE blow-uprate' Type I condition. More precisely, we prove that if the vorticitydirection 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 aregular point. This result is inspired by and improves the regularity criteriagiven by Giga, Hsu and Maekawa (2014). We also obtain new local versions forsuitable weak solutions near the flat boundary. Our method hinges on new scaledMorrey estimates, blow-up and compactness arguments and `persistence ofsingularites' on the flat boundary. The scaled Morrey estimates seem to be ofindependent interest.

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