On the $$C_0$$ semigroup generated by the Oseen operator around a steady flow exterior to a rotating obstacle

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We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $$C_0$$ -semigroup in $$L^p$$ space and then develop a series of $$L^p$$ – $$L^q$$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $$L^3$$ and the steady flow are sufficiently small.