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We obtain an improved blow-up criterion for solutions of the Navier-Stokesequations in critical Besov spaces. If a mild solution $u$ has maximalexistence time $T^* < \infty$, then the non-endpoint critical Besov norms mustbecome infinite at the blow-up time: $$ \lim_{t \uparrow T^*}\lVert{u(\cdot,t)}\rVert_{\dot B^{-1+3/p}_{p,q}(\mathbb{R}^3)} = \infty, \quad3 < p,q < \infty. $$ In particular, we introduce a priori estimates for thesolution based on elementary splittings of initial data in critical Besovspaces and energy methods. These estimates allow us to rescale around apotential singularity and apply backward uniqueness arguments. The proof doesnot use profile decomposition.

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