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

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.

Authors and Affiliations

1. Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano 25/B, I-56126 Pisa, Italy, e-mail: bveiga@dma.unipi.it, , , , , , IT

   H. Beirao da Veiga

Accepted: April 21, 2000

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