Remarks on Interior Regularity Criteria Without Pressure for the Navier-Stokes Equations

Abstract

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the $L_t^p{L}_x^q(Q_1)$ norm of the velocity is sufficiently small, where $1\le \frac{2}{p}+\frac{3}{q}<2$ and $2\le p\le \infty$. It improves the recent result of $p,q>2$ by Kwon \cite{Kwon} (J. Differential Equations 357 (2023), 1--31.), and also generalizes Chae-Wolf's $L_t^{\infty }{L}_x^{\frac{3}{2}+}$ criterion \cite{CW2017} (Arch. Ration. Mech. Anal. 225 (2017), no. 1, 549--572.).