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

Albritton, Dallas

doi:10.2140/apde.2018.11.1415

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

Dallas Albritton

Vol. 11 (2018), No. 6, 1415–1456

DOI: 10.2140/apde.2018.11.1415

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 $T^{*} < \infty$ , then the non-endpoint critical Besov norms must become infinite at the blow-up time:

lim_{t ↑ T^{*}} ∥ u (\cdot, t) ∥_{Ḃ_{p, q}^{- 1 + 3 ∕ p} (ℝ^{3})} = \infty, 3 < p, q < \infty .

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.

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Keywords

Navier–Stokes equations, Besov spaces, blow-up criteria

Mathematical Subject Classification 2010

Primary: 35Q30

Milestones

Received: 20 February 2017

Revised: 3 December 2017

Accepted: 14 February 2018

Published: 3 May 2018

Authors

Dallas Albritton
School of Mathematics University of Minnesota Minneapolis, MN United States

Vol. 11, No. 6, 2018