On the Smoothness of a Class of Weak Solutions to the Navier—Stokes equations

Abstract.

We improve regularity criteria for weak solutions to the Navier-Stokes equations stated in references [1], [3] and [12], by using in the proof given in [3], a new idea introduced by H. O. Bae and H. J. Choe in [1]. This idea allows us, in one of the main hypothesis (see eq. (1.7)), to replace the velocity u by its projection $\bar u$ into an arbitrary hyperplane of ${\Bbb R}^n$; see Theorem A. For simplicity, we state our results for space dimension $n \le 4$, since if $n \ge 5$ the proofs become more technical and additional hypotheses are needed. However, for the interested reader, we will present the formal calculations for arbitrary dimension n.