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

Abstract

We show that if $v$ is a smooth suitable weak solution to the Navier-Stokes equations on $B(0,4)\times (0,T_{\ast })$, that possesses a singular point $(x_0,T_{\ast })\in B(0,4)\times \{T_{\ast }\}$, then for all $\delta >0$ sufficiently small one necessarily has $$\lim \underset{t\uparrow T_{\ast }}{\sup }\frac{\Vert v(\cdot ,t){\Vert }_{L^3(B(x_0,\delta ))}}{\left(\log \log \log \right(\frac{1}{(T_{\ast }-t{)}^{\frac{1}{4}}}){)}^{\frac{1}{1129}}}=\infty .$$ 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.

36 pages