LAUSR

Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions, 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 $$L^q$$ L q generated by the linearized non-autonomous system was proved by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) and the large time behavior of T ( t ,  s ) f in $$L^r$$ L r for $$(t-s)\rightarrow \infty $$ ( t - s ) → ∞ was then developed by Hishida (Math Ann 372:915–949, 2018) when f is taken from $$L^q$$ L q with $$q\leqq r$$ q ≦ r . The contribution of the present paper concerns such $$L^q$$ L q - $$L^r$$ L r decay estimates of $$\nabla T(t,s)$$ ∇ 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 q - $$L^r$$ L r estimates for the autonomous case (Stokes and Oseen semigroups, those semigroups with rotating effect) in three dimensional exterior domains, which were established by Hishida and Shibata (Arch Ration Mech Anal 193:339–421, 2009), Iwashita (Math Ann 285, 265–288, 1989), Kobayashi and Shibata (Math Ann 310:1–45, 1998), Maremonti and Solonnikov (Ann Sc Norm Super Pisa 24:395–449, 1997) and Shibata (in: Amann, Arendt, Hieber, Neubrander, Nicaise, von Below (eds) Functional analysis and evolution equations, the Günter Lumer volume. Birkhäuser, Basel, pp 595–611, 2008).