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

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 $\Vert r^{1-\frac{3}{q}}u{\Vert }_{L_t^{\infty }{L}_x^q}<\infty$ 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.

The final version to appear in ARMA. 48 pages, 1 figure