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A Beale–Kato–Majda Criterion with Optimal Frequency and Temporal Localization

  • Xiaoyutao LuoEmail author
Article

Abstract

We obtain a Beale–Kato–Majda-type criterion with optimal frequency and temporal localization for the 3D Navier–Stokes equations. Compared to previous results our condition only requires the control of Fourier modes below a critical frequency, whose value is explicit in terms of time scales. As applications it yields a strongly frequency-localized condition for regularity in the space \(B^{-1}_{\infty ,\infty }\) and also a lower bound on the decaying rate of \(L^p\) norms \(2\le p <3\) for possible blowup solutions. The proof relies on new estimates for the cutoff dissipation and energy at small time scales which might be of independent interest.

Notes

Acknowledgements

The author would like to thank his advisor Alexey Cheskidov for stimulating conversations and his constant encouragement. The author was partially supported by the NSF Grant DMS 1517583 through his advisor Alexey Cheskidov. The author also thanks anonymous referees for carefully reading the manuscript and giving many useful comments and suggestions, which greatly improve the quality of this work.

Compliance with Ethical Standards

Conflict of interest

The author declares that he has no conflict of interest.

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Authors and Affiliations

  1. 1.Department of Mathematics Statistics and Computer ScienceUniversity of Illinois At ChicagoChicagoUSA

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