Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains

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].