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

$\underset{t↑{T}^{\ast }}{lim}\parallel u\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}},t\right){\parallel }_{{Ḃ}_{p,q}^{-1+3∕p}\left({ℝ}^{3}\right)}=\infty ,\phantom{\rule{1em}{0ex}}3

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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