Local energy bounds and \(\epsilon \)-regularity criteria for the 3D Navier–Stokes system

Abstract

The system of three dimensional Navier–Stokes equations is considered. We obtain some new local energy bounds that enable us to improve several \(\epsilon \)-regularity criteria. The key idea here is to view the ‘head pressure’ as a signed distribution belonging to certain fractional Sobolev space of negative order. This allows us to capture the oscillation of the pressure in our criteria.

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Acknowledgements

The second author is supported in part by Simons Foundation, award number 426071. He also wishes to acknowledge the support from Institut des Hautes Études Scientifiques (France), where part of this work was done. The authors graciously thank the anonymous referee for valuable comments that help improve the quality of the paper. In particular, the short proof of Lemma 4.1 was generously suggested by the referee.

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Correspondence to Cristi Guevara.

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Communicated by M.Struwe.

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Guevara, C., Phuc, N.C. Local energy bounds and \(\epsilon \)-regularity criteria for the 3D Navier–Stokes system. Calc. Var. 56, 68 (2017). https://doi.org/10.1007/s00526-017-1151-7

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Mathematics Subject Classification

  • Primary 35Q30
  • Secondary 35Q35