  The goal of this paper is to provide an algorithm that, for any sufficiently
localised, divergence-free small initial data, explicitly constructs a
localised external force leading to a rapidly dissipative solutions of the
Navier-Stokes equations $\mathbb{R}^n$: namely, the energy decay rate of the
flow will be forced to satisfy $\|u(t)\|_2^2 = o(t^{-(n+2)/2})$ as $t \to
\infty$, which is beyond the usual optimal rate. An important feature of our
construction is that this force can always be taken compactly supported in
space-time, and its profile arbitrarily prescribed up to a spatial rescaling.
Since the forcing term vanishes after a finite time interval, our result
suggests that nontrivial interactions between the linear and nonlinear parts
occur, annihilating all the slowly decaying terms contained in Miyakawa and
Schonbek's asymptotic profiles.
