Quantitative partial regularity of the Navier-Stokes equations and applications

We prove a logarithmic improvement of the CaffarelliKohn-Nirenb erg partial regularity theorem for the NavierStokes equations. The key idea is to find a quantitative counterpart for the absolute continuity of the dissipation energy using a pigeonholing argument. Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending exponentially on the natural local energies. Then, we give two applications of the latter result in the axially symmetric case. The first one is a local regularity criterion for suitable weak solutions with small swirl. The second one is a slightly improved one-point CKN criterion which implies all known (slightly supercritical) Type I regularity results in the literature. (c) 2024 Elsevier Inc. All rights reserved.