In this paper we prove the regularity of Leray weak solutions of the Navier–Stokes equations as long as the vorticity projection to a plane is bounded in the scale critical space \(L^p(0,T;L^q)\), \(2/p+3/q=2\), \(q \in (3/2,\infty )\). The plane may vary in space and time while the unit vector \(v=v(x,t)\) orthogonal to the plane is locally a Hölder function in space with the coefficient 1/2. This extends previous works by Chae and Choe and by Miller. We further show that a generalized form of this criterion improves several other regularity criteria in terms of the vorticity direction known from the literature.

本文证明了Navier-Stokes方程的Leray弱解的正则性，只要对平面的涡旋投影在尺度临界空间\（L ^ p（0，T; L ^ q）\）内，\（2 / p + 3 / q = 2 \），\（q \ in（3/2，\ infty）\）。平面在空间和时间上可能会发生变化，而与平面正交的单位矢量\（v = v（x，t）\）在局部空间上是Hölder函数，系数为1/2。这是蔡和崔以及米勒先前作品的延伸。我们进一步表明，根据文献中已知的涡度方向，该准则的广义形式可以改善其他几个规则性准则。