Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in
Section snippets
Introduction and main result
We are concerned with the following Cauchy problem for the 3D Navier–Stokes equations: where and are unknown velocity and pressure, respectively, and is a given initial velocity.
It was proved in [1] that for , the problem (1.1) has a global weak solution (Leray–Hopf weak solution) which satisfies (1.1) in the sense of distributions and
Preliminaries
Throughout the paper we will use the following notations. We denote by the positive constants which may vary from line to line. For simplicity, we omit in function spaces defined on as long as no confusion arises. For a normed space , we denote by the -norm. In particular, if , then denotes the -norm. We denote by and Bessel potential space and homogeneous Sobolev spaces, respectively (cf. [26]).
Let be the space of tempered distributions. and
Proof of the main results
In this section we prove Theorem 1.1, Theorem 1.2. For the proof of the main results, we establish the following trilinear estimate involving -norm, which is proved using the Bony decomposition and (2.9).
Lemma 3.1 Let be such that . Then there exists an absolute constant such that for any and .
By (1.19) we find that the case when in Lemma 3.1 gives the sharper estimate than the case when in Lemma 5 of [20].
Proof of Lemma 3.1 By
Acknowledgements
The author would like to express his sincere thanks to the editor and the anonymous referees for their valuable suggestions concerning the paper.
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