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

TL;DR

This paper establishes explicit bounds and criteria for the regularity and potential blow-up of solutions to the 3D Navier-Stokes equations within Lorentz spaces, enhancing understanding of solution behavior in critical function spaces.

Contribution

It introduces a quantitative regularity theorem and blow-up rate in Lorentz spaces, extending previous Lebesgue space results and providing explicit bounds for classical solutions.

Findings

Derived explicit blow-up rates in Lorentz spaces

Improved regularity criteria over previous Lebesgue space results

Quantified the qualitative blow-up behavior in critical 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 \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].