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The Regularity of Weak Solutions of the 3D Navier–Stokes Equations in \({B^{-1}_{\infty,\infty}}\)

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Abstract

We show that if a Leray–Hopf solution u of the three-dimensional Navier–Stokes equation belongs to \({C((0,T]; B^{-1}_{\infty,\infty})}\) or its jumps in the \({B^{-1}_{\infty,\infty}}\)-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya–Prodi–Serrin criterion.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    A. Cheskidov

  2. Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL, 60607, USA

    R. Shvydkoy

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  1. A. Cheskidov
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  2. R. Shvydkoy
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Correspondence to R. Shvydkoy.

Additional information

Communicated by V. Sverak

The work of R. Shvydkoy was partially supported by NSF grant DMS-0604050.

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Cheskidov, A., Shvydkoy, R. The Regularity of Weak Solutions of the 3D Navier–Stokes Equations in \({B^{-1}_{\infty,\infty}}\) . Arch Rational Mech Anal 195, 159–169 (2010). https://doi.org/10.1007/s00205-009-0265-2

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  • DOI: https://doi.org/10.1007/s00205-009-0265-2

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