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

TL;DR

This paper establishes an improved blow-up criterion for Navier-Stokes solutions in critical Besov spaces, showing that certain norms must become infinite at the blow-up time, and introduces new a priori estimates without profile decomposition.

Contribution

It provides a new blow-up criterion in non-endpoint critical Besov spaces and develops novel a priori estimates using elementary splittings and energy methods.

Findings

Blow-up occurs only if critical Besov norms become infinite.

Introduces a priori estimates based on initial data splittings.

Proof avoids using profile decomposition techniques.

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: $$ \lim_{t \uparrow T^*} \lVert{u(\cdot,t)}\rVert_{\dot B^{-1+3/p}_{p,q}(\mathbb{R}^3)} = \infty, \quad 3 < p,q < \infty. $$ 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.