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

2022/09/30 by Barker, Tobias · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.15627

Abstract

We show that if v is a smooth suitable weak solution to the Navier-Stokes equations on B(0,4)× (0,T_*), that possesses a singular point (x₀,T_*)∈ B(0,4)× \T_*\, then for all δ>0 sufficiently small one necessarily has limsup_{t\uparrow T_*} \frac‖v(⋅,t)‖_{L³(B(x0,δ))}(logloglog(\frac1(T_*-t)^(1)/(4)))^(1)/(1129)=∞. 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.

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