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

TL;DR

The authors address quantitative regularity and blow-up behavior for the 3D Navier–Stokes equations in critical Lorentz spaces $L^{3,q_0}$ by adapting Tao's strategy to this setting. They construct a Lorentz-space–adapted toolkit, including Littlewood–Paley analysis, Lorentz-Hölder/Young inequalities, and heat-kernel bounds, to derive a precise blow-up rate and a sharp regularity criterion in $L^{3,q_0}$. Their main contributions include a quantitative blow-up rate expressed in terms of the $L^{3,q_0}$ norm and a robust a priori framework that yields control of derivatives under a bounded Lorentz norm, extending and refining prior $L^3$ results and Lorentz-space criteria through Carleman estimates and a cascade of regularity regions. These results advance understanding of singularity formation in the NSE and provide scale-explicit, quantitative criteria in critical Lorentz spaces with potential implications for regularity theory and fluid dynamics analyses.

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].

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Remark 2
• Definition 2.1
• Lemma 2.2: Hölder's inequality, hunt1966p
• Lemma 2.3: Young's inequality, o1963convolution
• Lemma 2.4: Sobolev's inequality, tartar1998imbedding
• Lemma 2.5: Bernstein inequality
• proof