Quantitative bounds for critically bounded solutions to the three-dimensional Navier-Stokes equations in Lorentz spaces

Abstract

In this paper, we prove a quantitative regularity theorem and a blow-up criterion of classical solutions for the three-dimensional Navier-Stokes equations. By adapting the strategy developed by Tao in [20], we obtain an explicit blow-up rate in the setting of critical Lorentz spaces $L^{3,q_0}({\mathbb{R}}^3)$ with $3\le q_0<\infty$. Our results improve the previous regularity in critical Lebesgue spaces $L^3({\mathbb{R}}^3)$ in [20] and quantify the qualitative result by Phuc in [16].