Communications in Mathematical Physics ( IF 2.4 ) Pub Date : 2021-06-14 , DOI: 10.1007/s00220-021-04122-x Tobias Barker , Christophe Prange
This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound \(\Vert u\Vert _{L^{\infty }_{t}L^{3,\infty }_{x}}\le M\). Namely, we show that if \(T^*\) is a first blow-up time and \((0,T^*)\) is a singular point then
$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{0}(R))}\ge C(M)\log \Big (\frac{1}{T^*-t}\Big ),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}). \end{aligned}$$We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on \({\mathbb {R}}^3\times (0,1)\) with
$$\begin{aligned} \sup _{n}\Vert u(\cdot ,t_{(n)})\Vert _{L^{3}({\mathbb {R}}^3)}<\infty \,\,\,\text {and}\,\,\,t_{(n)}\uparrow 1, \end{aligned}$$then u does not blow-up at \(t=1\). To prove our results we develop a new strategy for proving quantitative bounds for the Navier–Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.
中文翻译：
纳维-斯托克斯方程通过空间浓度的定量规律
本文关注的是 Navier-Stokes 方程的定量估计。首先，我们研究定量边界与 I 类边界附近的临界规范行为的关系\(\Vert u\Vert _{L^{\infty }_{t}L^{3,\infty }_ {x}}\le M\)。也就是说，我们证明如果\(T^*\)是第一次爆炸时间并且\((0,T^*)\)是一个奇异点，那么
$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{0}(R))}\ge C(M)\log \Big (\frac{1 {T^*-t}\Big ),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-})。\end{对齐}$$我们证明了这种潜在的膨胀率对于某类潜在的非零后向离散自相似解是最佳的。其次，我们量化了 Seregin 的结果（Commun Math Phys 312(3):833–845, 2012），它表示如果u是\({\mathbb {R }}^3\times (0,1)\)与
$$\begin{aligned} \sup _{n}\Vert u(\cdot ,t_{(n)})\Vert _{L^{3}({\mathbb {R}}^3)}<\ infty \,\,\,\text {and}\,\,\,t_{(n)}\uparrow 1, \end{aligned}$$那么你不会在\(t=1\)处爆炸。为了证明我们的结果，我们开发了一种新策略来证明 Navier-Stokes 方程的定量界限。这取决于 Jia 和 Šverák（2014 年）建立的空间局部平滑结果（接近初始时间），以及 Tao（2019 年）给出的使用卡尔曼不等式的定量论证。此外，这里开发的技术特别使我们能够在 I 型爆炸场景中给出奇异点数量的定量界限。
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