  In this paper we consider classes of initial data that ensure local-in-time
Hadamard well-posedness of the associated weak Leray-Hopf solutions of the
three-dimensional Navier-Stokes equations. In particular, for any solenodial
$L_{2}$ initial data $u_{0}$ belonging to certain subsets of
$VMO^{-1}(\mathbb{R}^3)$, we show that weak Leray-Hopf solutions depend
continuously with respect to small divergence-free $L_{2}$ perturbations of the
initial data $u_{0}$ (on some finite-time interval). Our main result is
inspired and improves upon previous work of the author \cite{barker2018} and
work of Jean-Yves Chemin \cite{chemin}. Our method builds upon
\cite{barker2018} and \cite{chemin}. In particular our method hinges on
decomposition results for the initial data inspired by Calder\'{o}n
\cite{Calderon90} together with use of persistence of regularity results. The
persistence of regularity statement presented may be of independent interest,
since it does not rely upon the solution or the initial data being in the
perturbative regime.
