Abstract
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
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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Acknowledgements
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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Han, B., Lei, Z., Li, D. et al. Sharp One Component Regularity for Navier–Stokes. Arch Rational Mech Anal 231, 939–970 (2019). https://doi.org/10.1007/s00205-018-1292-7
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DOI: https://doi.org/10.1007/s00205-018-1292-7