In this paper, we present a regularity criterion to the Navier–Stokes equations based on one entry of the velocity gradient. In fact, we prove that the weak solution to the Navier-Stokes equations is regular provided that \(\partial _{3}u_{3}\in L^{\beta }(0, T; L^{\alpha }(\mathbb {R} ^{3}))\) with \(\alpha >\frac{7+\sqrt{13}}{6}\) and: \(\begin{aligned} \frac{2}{\beta }+\frac{3}{\alpha }= \frac{-12\,\widehat{\alpha }^{2}+28\, \widehat{\alpha }-3+\sqrt{ (3-2\, \widehat{\alpha })(-72\,\widehat{\alpha }^{3}+276\, \widehat{\alpha }^{2}-374\,\widehat{\alpha }+195)}}{8(2-\widehat{\alpha })}, \end{aligned}\) where \( \widehat{\alpha }=\frac{1}{\alpha }.\) This result improves the previous result obtained by Zujin Zhang and Yali Zhang in (Z. Angew. Math. Phys.)(2021), which states the similar result for \(\alpha \ge \frac{3+\sqrt{17}}{4}.\) Notice that \(\frac{7+\sqrt{13}}{6}<\frac{3+\sqrt{17}}{4},\) thus the range of \(\alpha \) is changed. Also, we show that \(\beta \) corresponding to \(\alpha \) which is obtained in our result is smaller than \(\beta \) obtained in the mentioned paper.
