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

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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 ) p , q 1 + 3 p ( 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
Received: 20 February 2017
Revised: 3 December 2017
Accepted: 14 February 2018
First available in Project Euclid: 23 May 2018

Permanent link to this document
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
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Keywords
Navier–Stokes equations Besov spaces blow-up criteria

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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