In this paper we consider the regularity problem of the Navier–Stokes equations in \( {\mathbb {R}}^{3} \). We show that the Serrin-type condition imposed on one component of the velocity \( u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))\) with \( \frac{2}{p}+ \frac{3}{q} <1\), \( 3<q \le +\infty \) implies the regularity of the weak Leray solution \( u: {\mathbb {R}}^{3} \times (0,T) \rightarrow {\mathbb {R}}^{3} \), with the initial data belonging to \( L^2({\mathbb {R}}^3) \cap L^3({\mathbb {R}}^{3})\). The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

在本文中，我们考虑了\（{\ mathbb {R}} ^ {3} \）中Navier–Stokes方程的正则性问题。我们证明了Serrin型条件强加于速度\（u_3 \ in L ^ p（0，T; L ^ q（{\ mathbb {R}} ^ {3}））\）中的\（ \ frac {2} {p} + \ frac {3} {q} <1 \），\（3 <q \ le + \ infty \）表示弱Leray解的正则性\（u：{\ mathbb { R}} ^ {3} \ times（0，T）\ rightarrow {\ mathbb {R}} ^ {3} \），其初始数据属于\（L ^ 2（{\ mathbb {R}} ^ 3）\ cap L ^ 3（{\ mathbb {R}} ^ {3}）\）。该结果是针对合适的弱解的一个速度分量方面的新局部规则性准则的直接结果。