## Analysis & PDE

• Anal. PDE
• Volume 11, Number 6 (2018), 1415-1456.

### Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces

Dallas Albritton

#### Abstract

We obtain an improved blow-up criterion for solutions of the Navier–Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T ∗ < ∞$, then the non-endpoint critical Besov norms must become infinite at the blow-up time:

$lim t ↑ T ∗ ∥ u ( ⋅ , t ) ∥ Ḃ p , q − 1 + 3 ∕ p ( ℝ 3 ) = ∞ , 3 < p , q < ∞ .$

In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

#### Article information

Source
Anal. PDE, Volume 11, Number 6 (2018), 1415-1456.

Dates
Revised: 3 December 2017
Accepted: 14 February 2018
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.apde/1527040816

Digital Object Identifier
doi:10.2140/apde.2018.11.1415

Mathematical Reviews number (MathSciNet)
MR3803715

Zentralblatt MATH identifier
06881247

Subjects

#### Citation

Albritton, Dallas. Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces. Anal. PDE 11 (2018), no. 6, 1415--1456. doi:10.2140/apde.2018.11.1415. https://projecteuclid.org/euclid.apde/1527040816

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