# The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and 3D NSE

The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and... In the context of the $${L^\infty}$$ L ∞ -theory of the 3D NSE, it is shown that smallness of a solution in Besov space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 suffices to prevent a possible blow-up. In particular, it is revealed that the aforementioned condition implies a particular local spatial structure of the regions of high velocity magnitude, namely, the structure of local volumetric sparseness on the scale comparable to the radius of spatial analyticity measured in $${L^\infty}$$ L ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# The Space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 , Volumetric Sparseness, and 3D NSE

, Volume 19 (3) – Sep 6, 2016
9 pages

/lp/springer_journal/the-space-b-1-infty-infty-b-1-volumetric-sparseness-and-3d-nse-oNs5kwYdb7
Publisher
Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-016-0288-z
Publisher site
See Article on Publisher Site

### Abstract

In the context of the $${L^\infty}$$ L ∞ -theory of the 3D NSE, it is shown that smallness of a solution in Besov space $${B^{-1}_{\infty, \infty}}$$ B ∞ , ∞ - 1 suffices to prevent a possible blow-up. In particular, it is revealed that the aforementioned condition implies a particular local spatial structure of the regions of high velocity magnitude, namely, the structure of local volumetric sparseness on the scale comparable to the radius of spatial analyticity measured in $${L^\infty}$$ L ∞ .

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Sep 6, 2016

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