  We introduce a notion of global weak solution to the Navier-Stokes equations
in three dimensions with initial values in the critical homogeneous Besov
spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy
a certain stability property with respect to the weak-$\ast$ convergence of
initial conditions. To illustrate this property, we provide applications to
blow-up criteria, minimal blow-up initial data, and forward self-similar
solutions. Our proof relies on a new splitting result in homogeneous Besov
spaces that may be of independent interest.
