  In 2016, Seregin and \u{S}ver\'ak, conceived a notion of global in time
solution (as well as proving existence of them) to the three dimensional
Navier-Stokes equation with $L_3$ solenoidal initial data called 'global $L_3$
solutions'. A key feature of global $L_3$ solutions is continuity with respect
to weak convergence of a sequence of solenoidal $L_3$ initial data. The first
aim of this paper is to show that a similar notion of ' global
$\dot{B}^{-\frac{1}{4}}_{4,\infty}$ solutions' exists for solenoidal initial
data in the wider critical space $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ and
satisfies certain continuity properties with respect to weak* convergence of a
sequence of solenoidal $\dot{B}^{-\frac{1}{4}}_{4,\infty}$ initial data. This
is the widest such critical space if one requires the solution to the
Navier-Stokes equations minus the caloric extension of the initial data to be
in the global energy class.
  For the case of initial values in the wider class of
$\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$ initial data ($p>4)$, we prove that for
any $0<T<\infty$ there exists a solution to the Navier-Stokes system on
$\mathbb{R}^3 \times ]0,T[$ with this initial data. We discuss how properties
of these solutions imply a new regularity criteria for 3D weak Leray-Hopf
solutions in terms of the norm
$\|v(\cdot,t)\|_{\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}$ (as well as certain
additional assumptions).
  The main new observation of this paper, that enables these results, regards
the decomposition of homogeneous Besov spaces $\dot{B}^{-1+\frac 3
p}_{p,\infty}$. This does not appear to obviously follow from the known real
interpolation theory.
