Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces

TL;DR

This paper establishes a quantitative regularity theorem and blowup criterion for 3D Navier-Stokes solutions in critical spaces, improving estimates and providing new bounds on blowup rates using advanced analytical tools.

Contribution

It introduces improved subcritical estimates for solutions in critical spaces, extending Tao's strategy, and derives a double logarithmic lower bound on blowup rates.

Findings

Double exponential bounds on solution norms

Double logarithmic lower bound on blowup rate

Enhanced regularity criteria for critical solutions

Abstract

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.