  There is considerable evidence that solutions to the non-forced 3D
Navier-Stokes equations in the natural energy space are not unique. Assuming
this is the case, it becomes important to quantify how non-uniqueness evolves.
In this paper we provide an algebraic estimate for how rapidly two possibly
non-unique solutions can separate over a compact spatial region in which the
initial data has sub-critical regularity. Outside of this compact region, the
data is only assumed to be in the scaling critical weak Lebesgue space and can
be large. In order to establish this separation rate, we develop a new
spatially local, short-time asymptotic expansion which is of independent
interest.
