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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 3 ∈ L ∞(0, T; Ḃ p,q −1+3/p (ℝ3)) for 3 < p,q < ∞ and u h ∈ L ∞(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; Ḃ 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.

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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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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    WenDong Wang

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    ZhiFei Zhang

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  1. WenDong Wang
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  2. ZhiFei Zhang
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Correspondence to WenDong Wang.

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Wang, W., Zhang, Z. Blow-up of critical norms for the 3-D Navier-Stokes equations. Sci. China Math. 60, 637–650 (2017). https://doi.org/10.1007/s11425-016-0344-5

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