# Lower Bounds for Possible Singular Solutions for the Navier–Stokes and Euler Equations Revisited

Lower Bounds for Possible Singular Solutions for the Navier–Stokes and Euler Equations Revisited In this paper we give optimal lower bounds for the blow-up rate of the $$\dot{H}^{s}\left( \mathbb {T}^3\right)$$ H ˙ s T 3 -norm, $$\frac{1}{2}<s<\frac{5}{2}$$ 1 2 < s < 5 2 , of a putative singular solution of the Navier–Stokes equations, and we also present an elementary proof for a lower bound on blow-up rate of the Sobolev norms of possible singular solutions to the Euler equations when $$s>\frac{5}{2}$$ s > 5 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# Lower Bounds for Possible Singular Solutions for the Navier–Stokes and Euler Equations Revisited

, Volume 20 (1) – Nov 30, 2016
5 pages

/lp/springer_journal/lower-bounds-for-possible-singular-solutions-for-the-navier-stokes-and-aI2I24I8m0
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/s00021-016-0308-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper we give optimal lower bounds for the blow-up rate of the $$\dot{H}^{s}\left( \mathbb {T}^3\right)$$ H ˙ s T 3 -norm, $$\frac{1}{2}<s<\frac{5}{2}$$ 1 2 < s < 5 2 , of a putative singular solution of the Navier–Stokes equations, and we also present an elementary proof for a lower bound on blow-up rate of the Sobolev norms of possible singular solutions to the Euler equations when $$s>\frac{5}{2}$$ s > 5 2 .

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Nov 30, 2016

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