论文标题
Navier-Stokes方程的定量规律性通过空间浓度
Quantitative regularity for the Navier-Stokes equations via spatial concentration
论文作者
论文摘要
本文涉及Navier-Stokes方程的定量估计。 首先，我们研究了定量界限与潜在奇点附近的关键规范的行为的关系，I型I绑定$ \ | U \ | _ {l^{\ infty} _ {t} _ {t} l^{3，\ infty} _ {x}}}}}}} \ leq m $。即，我们表明，如果$ t^*$是第一个爆破时间，而$（0，t^*）$是一个单数点，则$$ \ | u（\ cdot，t）\ | _ {l^{3}（b_ {0}（b_ {0}（r））} \ geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar解决方案。 其次，我们量化了Seregin（2012）的结果，该结果说，如果$ u $是$ \ m athbb {r}^3 \ times（0,1）$的Navier-Stokes方程的平滑有限能力解决方案$ \ sup_ {n} \ | u（\ cdot，t _ {（n）}）\ | _ {l^{3}（\ Mathbb {r}^3）} <\ infty \，\，\，\，\，\，\，\，\，\，\，\ textrm {和textrm {and} {和\，\，\，\，\，\， $ u $在$ t = 1 $时不会爆炸。 为了证明我们的结果，我们制定了一种新的策略，以证明Navier-Stokes方程的定量界限。这取决于Jia和šverák（2014）建立的本地空间平滑结果（接近初始时间），以及使用Tao（2019）给出的Carleman不平等的定量论点。 此外，这里开发的技术使我们特别能够在I型爆破场景中对单数点的数量进行定量界限。
This paper is concerned with quantitative estimates for the Navier-Stokes equations. First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound $\|u\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\leq M$. Namely, we show that if $T^*$ is a first blow-up time and $(0,T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{0}(R))}\geq C(M)\log\Big(\frac{1}{T^*-t}\Big),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}).$$ 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 (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 $$\sup_{n}\|u(\cdot,t_{(n)})\|_{L^{3}(\mathbb{R}^3)}<\infty\,\,\,\textrm{and}\,\,\,t_{(n)}\uparrow 1,$$ 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.
