Science China Mathematics

, Volume 60, Issue 4, pp 637–650 | Cite as

Blow-up of critical norms for the 3-D Navier-Stokes equations

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Abstract

Let u = (u h, u 3) be a smooth solution of the 3-D Navier-Stokes equations in ℝ3 × [0, T). It was proved that if u 3L (0, T; p,q −1+3/p (ℝ3)) for 3 < p,q < ∞ and u hL (0, T; BMO−1(ℝ3)) with u h(T) ∈ VMO−1(ℝ3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al. (2016), which requires u ∈ L (0, T; p,q −1+3/p (ℝ3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.

Keywords

Navier-Stokes equations interior regularity criterion BMO space Besov space 

MSC(2010)

35Q30 76D05 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11301048, 11371039 and 11425103) and the Fundamental Research Funds for the Central Universities.

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina

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