Archive for Rational Mechanics and Analysis

, Volume 231, Issue 2, pp 939–970 | Cite as

Sharp One Component Regularity for Navier–Stokes

  • Bin Han
  • Zhen Lei
  • Dong Li
  • Na Zhao


We consider the conditional regularity of mild solution \({\nu}\) to the incompressible Navier–Stokes equations in three dimensions. Let \({e \in \mathbb{S}^{2}}\) and \({0 < {T}^{*} < \infty}\). Chemin and Zhang (Ann Sci Éc Norm Supér 49:131–167, 2016) proved the regularity of \({\nu}\) on (0, T*] if there exists \({p \in (4, 6)}\) such that
$$\int_{0}^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}} {\rm d}t < \infty. $$
Chemin et al. (Arch Ration Mech Anal 224(3):871–905, 2017) extended the range of p to \({(4,\infty)}\). In this article we settle the case \({p \in [2, 4]}\). Our proof also works for the case \({p \in (4,\infty)}\).


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The first author was in part supported by NSFC (No. 11701131) and Zhejiang Province Science fund forYouths (No. LQ17A010007). Liwas supported in part by Hong Kong RGC Grant GRF 16307317. Z. Lei and N. Zhao was in part supported by NSFC (Grant No. 11725102), National Support Program for Young Top-Notch Talents and SGST 09DZ2272900 from Shanghai Key Laboratory for Contemporary Applied Mathematics.

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Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Dianzi UniversityHangzhouChina
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina
  3. 3.Department of MathematicsThe Hong Kong University of Science and TechnologyKowloonHong Kong

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