Estimation of non-uniqueness and short-time asymptotic expansions for Navier-Stokes flows

There is considerable evidence that solutions to the three-dimensional 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. To establish this upper bound on the separation rate, we develop a new spatially local, short-time asymptotic expansion which is of independent interest.