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 \v{ S}ver\'{ a}k, 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 $$\|u(\cdot,t)\|_{ L^{ 3}(B_{ R}(0))}\geq \gamma_{ univ},\qquad R=O(\sqrt{ T^*-t}).$$ 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 $\dot B^{ -1+\frac3p}_{ p,\infty}$, $p\in(3,\infty)$.
Categories: math.AP