  Recent experimental and numerical observations have shown the significance of
the Basset--Boussinesq memory term on the dynamics of small spherical rigid
particles (or inertial particles) suspended in an ambient fluid flow. These
observations suggest an algebraic decay to an asymptotic state, as opposed to
the exponential convergence in the absence of the memory term. Here, we prove
that the observed algebraic decay is a universal property of the Maxey--Riley
equation. Specifically, the particle velocity decays algebraically in time to a
limit that is $\mathcal O(\epsilon)$-close to the fluid velocity, where
$0<\epsilon\ll 1$ is proportional to the square of the ratio of the particle
radius to the fluid characteristic length-scale. These results follows from a
sharp analytic upper bound that we derive for the particle velocity. For
completeness, we also present a first proof of existence and uniqueness of
global solutions to the Maxey--Riley equation, a nonlinear system of
fractional-order differential equations.
