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

We show that if v is a smooth suitable weak solution to the Navier-Stokes equations on B(0, 4) \times (0,T*), which possesses a singular point (x0,T*) \in B(0, 4) \times \{ T*\} , then for all \delta > 0 sufficiently small, one necessarily has limsupt↓T*| v( ,t)\| L3(B(x0,δ )) log log log( 1 (T*t) 1 4 )) 1 1129 = \infty . This local result improves on the corresponding global result recently established by Tao [in Nine Mathematical Challenges: An Elucidation, American Mathematical Society, Providence, RI, 2021, pp. 149-193]. The proof is based on a quantification of the qualitative local result of Escauriaza, Seregin, and \v Sverak [Uspekhi Mat. Nauk, 58 (2003), pp. 3-44]. 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 [in Applied Nonlinear Analysis, Kluwer/Plenum, New York, 1999, pp. 391-402]. After performing the quantitative truncation procedure, the remainder of the proof hinges on a physical space analogue of Tao's breakthrough strategy, established by Barker and Prange.