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.
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