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

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:

$$\underset{t\uparrow T^{\ast }}{lim}\parallel u\left(\cdot ,t\right){\parallel }_{{Ḃ}_{p,q}^{-1+3∕p}\left({ℝ}^3\right)}=\infty ,3<p,q<\infty .$$

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.