We show that the spatial norm in any critical homogeneous Besov
space in which local existence of strong solutions to the 3-d
Navier-Stokes equations is known must become unbounded near a singularity.
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. This is joint work with I. Gallagher (Paris 7) and
F. Planchon (Nice).