  Let us consider the motion of a viscous incompressible fluid past a rotating
rigid body in 3D, where the translational and angular velocities of the body
are prescribed but time-dependent. In a reference frame attached to the body,
we have the Navier-Stokes system with the drift and (one half of the) Coriolis
terms in a fixed exterior domain. The existence of the evolution operator
$T(t,s)$ in the space $L^q$ generated by the linearized non-autonomous system
was proved by Hansel and Rhandi [26] and the large time behavior of $T(t,s)f$
in $L^r$ for $(t-s)\to\infty$ was then developed by the present author [33]
when $f$ is taken from $L^q$ with $q\leq r$. The contribution of the present
paper concerns such $L^q$-$L^r$ decay estimates of $\nabla T(t,s)$ with optimal
rates, which must be useful for the study of stability/attainability of the
Navier-Stokes flow in several physically relevant situations. Our main theorem
completely recovers the $L^q$-$L^r$ estimates for the autonomous case (Stokes
and Oseen semigroups, those semigroups with rotating effect) in 3D exterior
domains, which were established by [37], [42], [39], [36] and [44].
