This paper is concerned with two dual aspects of the regularity question for 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 $$L^3$$ 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 $$(0, T^*)$$ ( 0 , T ∗ ) is a singular point, then $$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{R}(0))}\geqq \gamma _{univ},\qquad R=O(\sqrt{T^*-t}). \end{aligned}$$ ‖ u ( · , t ) ‖ L 3 ( B R ( 0 ) ) ≧ γ univ , R = O ( 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 $$L^{3,\infty }$$ L 3 , ∞ and the Besov space $${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }$$ B ˙ p , ∞ - 1 + 3 p , $$p\in (3,\infty )$$ p ∈ ( 3 , ∞ ) .

中文翻译：

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Navier-Stokes 方程的局部平滑和奇点附近临界范数的集中

本文关注 Navier-Stokes 方程正则性问题的两个对偶方面。首先，我们证明了局部能量解决方案的局部时间局部平滑效应。更准确地说，如果限制到单位球的初始数据属于尺度临界空间 $$L^3$$L 3 ，那么解在空间上局部平滑一段时间，这是量化的。这建立在考虑亚临界情况的 Jia 和 Šverák 的工作之上。其次，我们应用这些局部平滑估计来证明靠近可能的 I 型爆炸的集中现象。即，如果 $$(0, T^*)$$ ( 0 , T ∗ ) 是一个奇异点，则 $$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^ {3}(B_{R}(0))}\geqq \gamma _{univ},\qquad R=O(\sqrt{T^*-t})。\end{aligned}$$ ‖ u ( · , t ) ‖ L 3 ( BR ( 0 ) ) ≧ γ univ , R = O ( T ∗ - t ) 。该结果受到 Li、Ozawa、Wang 和 Maekawa、Miura 和 Prange 建立的专注结果的启发和改进。我们还将我们的结果扩展到其他临界空间，即 $$L^{3,\infty }$$ L 3 , ∞ 和 Besov 空间 $${\dot{B}}^{-1+\frac{3} {p}}_{p,\infty }$$ B ˙ p , ∞ - 1 + 3 p , $$p\in (3,\infty )$$ p ∈ ( 3 , ∞ ) 。