A regularity criterion for the Navier–Stokes equations via one diagonal entry of the velocity gradient

We study the conditional regularity of solutions to the Navier-Stokes equations in the three dimensional space. Let $u=(u_1,u_2,u_3)$ denote the velocity. We impose an additional condition only on one diagonal entry of the velocity gradient, namely $\partial_3 u_3$, and show, using a technique based on the mixed multiplier theorem and an anisotropic version of the Troisi inequality, that if $\partial_3 u_3$ lies in the space $L^\beta (0,T; L^q)$ with suitable $\beta,q$, then $u$ is regular on $(0,T]$. Our result improves and extends the analogous results known from the literature.

Keywords

Navier–Stokes equations, regularity criteria, mixed multiplier theorem, anisotropic version of Troisi inequality

2010 Mathematics Subject Classification

35Q30, 76D05

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The author was supported by the European Regional Development Fund, project No. CZ.02.1.01/0.0/0.0/16-019/0000778.

Received 5 November 2019

Accepted 17 December 2020

Published 18 June 2021