Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities

Abstract

This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $L^3$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Šverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if $(0, T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{R}(0))}\geq γ_{univ},\qquad R=O(\sqrt{T^*-t}).$$ This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely $L^{3,\infty}$ and the Besov space $\dot B^{-1+\frac3p}_{p,\infty}$, $p\in(3,\infty)$.