On Endpoint Regularity Criterion of the 3D Navier-Stokes equations

TL;DR

This paper establishes new endpoint regularity criteria for the 3D Navier-Stokes equations, showing smoothness under certain boundedness conditions in specific function spaces, improving previous results.

Contribution

The paper introduces novel endpoint regularity conditions involving Besov and Lorentz spaces, extending the criteria for smoothness of solutions to the 3D Navier-Stokes equations.

Findings

Proves smoothness of solutions under boundedness in $ ext{L}^ ext{infty}(0,T; ext{Besov})$ spaces.

Extends regularity criteria to include conditions on the third velocity component $u_3$.

Improves upon previous regularity results by Wang and Zhang.

Abstract

Let $(u, \pi)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $\mathbb{R}^3\times [0, T]$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C_0^\infty$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u\in L^\infty(0, T; \dot{B}^{-1}_{\infty,\infty})$, $u(x, T)\in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3\in L^\infty(0, T; L^{3, \infty})$ or $u_3\in L^\infty(0, T; \dot{B}^{-1+3/p}_{p, q})$ with $3<p, q< \infty$, then $u$ is smooth in $\mathbb{R}^3\times [0, T]$. Our result improves a previous result established by Wang and Zhang [Sci. China Math. 60, 637-650 (2017)].