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 \(B(0,4)\times (0,T_*)\) , which possesses a singular point \((x_0,T_*)\in B(0,4)\times \{T_*\}\) , then for all \(\delta \gt 0\) sufficiently small, one necessarily has \(\lim \sup_{t\uparrow T_*} \frac {\|v(\cdot,t)\|_{L^{3}(B(x_0,\delta ))}}{(\log \log \log (\frac {1}{(T_*-t)^{\frac {1}{4}}}))^{\frac {1}{1129}}}=\infty\) . 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 Šverak [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 [Comm. Math. Phys., 385 (2021), pp. 717–792].KeywordsNavier–Stokes equationslocal quantitative estimateslocal blow-up ratescritical normsMSC codes35Q3035K5535B4435B65

Locations

• arXiv (Cornell University) - View - PDF
• Pure (University of Bath) - View - PDF
• SIAM Journal on Mathematical Analysis - View