Localized quantitative estimates and potential blow-up rates for the Navier-Stokes equations

TL;DR

This paper establishes localized estimates and potential blow-up rates for solutions to the Navier-Stokes equations near singular points, refining previous global results and employing advanced quantification techniques.

Contribution

It provides the first localized quantitative estimates for Navier-Stokes singularities, improving upon recent global results and introducing new quantification methods.

Findings

Localized blow-up rate estimates near singular points.

Quantification of a qualitative truncation/localization procedure.

Extension of Tao's global blow-up results to local settings.

Abstract

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 $\delta>0$ sufficiently small one necessarily has $$\lim\sup_{t\uparrow T_*} \frac{\|v(\cdot,t)\|_{L^{3}(B(x_0,\delta))}}{\Big(\log\log\log\Big(\frac{1}{(T_*-t)^{\frac{1}{4}}}\Big)\Big)^{\frac{1}{1129}}}=\infty.$$ This local result improves upon the corresponding global result recently established by Tao. The proof is based upon a quantification of Escauriaza, Seregin and \v{S}verak'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.