We obtain an improved blow-up criterion for solutions of the Navier–Stokes equations in critical Besov spaces. If a mild solution has maximal existence time , then the non-endpoint critical Besov norms must become infinite at the blow-up time:
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.
"Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces." Anal. PDE 11 (6) 1415 - 1456, 2018. https://doi.org/10.2140/apde.2018.11.1415