  We are concerned with bilinear estimates and uniqueness of mild solutions for
the Navier-Stokes equations in critical spaces. For that, we construct general
settings in which estimates for the bilinear term of the mild formulation hold
true without using auxiliary norms such as Kato time-weighted ones. We first
obtain necessary conditions in abstract critical spaces and then consider
further structures to obtain the estimates in general classes of Besov, Morrey
and Besov-Morrey spaces based on Banach spaces. Examples of applications are
provided in different spaces as well as for other PDEs. In particular, as far
as we know, the bilinear estimate and the uniqueness property obtained in the
framework of Besov-weak-Herz spaces are not available in the existing
literature. The proofs are mainly based on characterizations and estimates on
the corresponding predual spaces.
