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- Title
On the Navier-Stokes equations in scaling-invariant spaces in any dimension.
- Authors
Kazuo Yamazaki
- Abstract
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.
- Subjects
NAVIER-Stokes equations; REARRANGEMENT invariant spaces; LAPLACIAN operator; BIOT-Savart law; LITTLEWOOD-Paley theory
- Publication
Revista Mathematica Iberoamericana, 2018, Vol 34, Issue 4, p1515
- ISSN
0213-2230
- Publication type
Article
- DOI
10.4171/rmi/1034