Blow-up of critical Besov norms at a potential Navier-Stokes singularity

We show that the spatial norm of any strong Navier-Stokes solution in the space X must become unbounded near a singularity, where X may be any critical homogeneous Besov space in which local existence of strong solutions to the 3-d Navier-Stokes system is known. In particular, the regularity of these spaces can be arbitrarily close to -1, which is the lowest regularity of any Navier-Stokes critical space. This extends a well-known result of Escauriaza-Seregin-Sverak (2003) concerning the Lebesgue space $L^3$, a critical space with regularity 0 which is continuously embedded into the spaces we consider. We follow the "critical element" reductio ad absurdum method of Kenig-Merle based on profile decompositions, but due to the low regularity of the spaces considered we rely on an iterative algorithm to improve low-regularity bounds on solutions to bounds on a part of the solution in spaces with positive regularity.