Vol. 11, No. 6, 2018

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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
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:

limtTu( ,t)p,q1+3p(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.

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