On the Navier-Stokes equations in scaling-invariant spaces in any dimension

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We study the Navier-Stokes equations with a dissipative term that is generalized through a fractional Laplacian in any dimension higher than two. We extend the horizontal Biot-Savart law beyond dimension three. Using the anisotropic Littlewood-Paley theory with which we distinguish the first two directions from the rest, we obtain a blow-up criteria for its solution in norms which are invariant under the rescaling of these equations. The proof goes through for the classical Navier-Stokes equations if dimension is three, four or five. We also give heuristics and partial results toward further improvement.

Original languageEnglish
Pages (from-to)1515-1540
Number of pages26
JournalActa Horticulturae Sinica
Issue number12
StatePublished - 2018


  • Anisotropic Littlewood-Paley theory
  • Blow-up
  • Navier-Stokes equations
  • Regularity

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