Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in BMO1

https://doi.org/10.1016/j.nonrwa.2020.103271Get rights and content

Abstract

In this paper, we provide a regularity criterion for 3D Navier–Stokes equations in terms of two vorticity components, which extends a recent result established by Guo, Kučera and Skalák (2018). More precisely, we prove that a unique local strong solution u to 3D Navier–Stokes equations does not blow up at time T provided only two components of vorticity belongs to L2(0,T;BMO1). We also prove that u can be smoothly extended beyond T, if the horizontal gradient of horizontal components of velocity u belong to L2(0,T;BMO1). The proof relies on the technique of the Bony decomposition and some Fourier multiplier theorems.

Section snippets

Introduction and main result

We are concerned with the following Cauchy problem for the 3D Navier–Stokes equations: utΔu+(u)u+p=0in(0,)×R3,u=0in(0,)×R3,u(0,x)=u0(x)inR3,where u=u(x,t)=(u1(x,t),u2(x,t),u3(x,t)) and p=p(x,t) are unknown velocity and pressure, respectively, and u0=u0(x) is a given initial velocity.

It was proved in [1] that for u0L2(R3)(u0=0), the problem (1.1) has a global weak solution (Leray–Hopf weak solution) uL(0,;L2(R3))L2(0,;H1(R3))which satisfies (1.1) in the sense of distributions and

Preliminaries

Throughout the paper we will use the following notations. We denote by C the positive constants which may vary from line to line. For simplicity, we omit R3 in function spaces X(R3) defined on R3 as long as no confusion arises. For a normed space X, we denote by X the X-norm. In particular, if X=Lr, then r denotes the Lr-norm. We denote by Hps and Ẇpm Bessel potential space and homogeneous Sobolev spaces, respectively (cf. [26]).

Let S be the space of tempered distributions. F and F1

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 BMO1-norm, which is proved using the Bony decomposition and (2.9).

Lemma 3.1

Let 1<q,r< be such that 1q+1r=1. Then there exists an absolute constant C>0 such that R3fghdxCfBMO1(gqhr+gqhr)for any fBMO1,gWq1 and hWr1.

By (1.19) we find that the case when q=r=2 in Lemma 3.1 gives the sharper estimate than the case when p=3 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.

References (30)

  • CheminJ.Y. et al.

    On the critical one component regularity for the 3D Navier–Stokes system: general case

    Arch. Ration. Mech. Anal.

    (2017)
  • CheskidovA. et al.

    The regularity of weak solutions of the 3D Navier–Stokes equations in B,1

    Arch. Ration. Mech. Anal.

    (2010)
  • EscauriazaL. et al.

    L3,-solutions of Navier–Stokes equations and backward uniqueness

    Russian Math. Surveys

    (2003)
  • KozonoH. et al.

    Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier–Stokes equations

    Math. Nachr.

    (2004)
  • BradshawZ. et al.

    Frequency localized regularity criteria for the 3D Navier–Stokes equations

    Arch. Ration. Mech. Anal.

    (2017)
  • Cited by (0)

    View full text