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

This paper is concerned with geometric regularity criteria for the Navier–Stokes equations in R3+×(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

⋃t∈(T−1,T)(B(0,R)∩R3+)×{t},R=O(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 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.