论文标题
Navier-Stokes方程的局部定量估计和潜在的爆炸率
Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations
论文作者
论文摘要
我们表明，如果$ v $是$ b（0,4）\ times（0，t _*）$上Navier-Stokes方程的平稳弱解决方案，则具有单个点$（x_0，t _*）\ in B（0,4）\ times \ times \ times \ times \ {t _***\} $，然后用$ fime $ fimession $ fimess \ times \ times \ times \ times \ times \ times \ time $ fimession。 $$ \ lim \ sup_ {t \ uparrow t_*} \ frac {\ | v（\ cdot，t）\ | _ {l^{3}（b（x_0，δ）}}}}} {\ big（\ log \ log \ log \ log \ log \ log \ big（\ fr ac {1} {（t _* - t）^{\ frac {1} {4}}}}}} \ big）\ big） Tao最近确定的相应全球结果改善了这种局部结果。证明是基于对雪茄菌，塞雷金和šverak的定性局部结果的量化。为了证明所需的局部定量估计，我们表明，在某些设置中，人们可以量化Neustupa和Penel引入的定性截断/定位程序。在执行定量截断程序之后，其余的证明取决于Prange和作者建立的Tao突破性策略的物理空间类似物。
We show that if $v$ is a smooth suitable weak solution to the Navier-Stokes equations on $B(0,4)\times (0,T_*)$, that possesses a singular point $(x_0,T_*)\in B(0,4)\times \{T_*\}$, then for all $δ>0$ sufficiently small one necessarily has $$\lim\sup_{t\uparrow T_*} \frac{\|v(\cdot,t)\|_{L^{3}(B(x_0,δ))}}{\Big(\log\log\log\Big(\frac{1}{(T_*-t)^{\frac{1}{4}}}\Big)\Big)^{\frac{1}{1129}}}=\infty.$$ This local result improves upon the corresponding global result recently established by Tao. The proof is based upon a quantification of Escauriaza, Seregin and Šverak's qualitative local result. In order to prove the required localized quantitative estimates, we show that in certain settings one can quantify a qualitative truncation/localization procedure introduced by Neustupa and Penel. After performing the quantitative truncation procedure, the remainder of the proof hinges on a physical space analogue of Tao's breakthrough strategy, established by Prange and the author.
