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

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 $3 \leq q_0 < \infty$. 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].