Boundedness and stability of solutions to the non-autonomous Oseen-Navier-Stokes equation ☆

We consider the motion of a viscous imcompressible fluid past a rotating rigid body in three-dimensional, where the translational and angular velocities of the body are prescribed but time-dependent. In a referenceframe attaches to the body, we have the non-autonomous Oseen-Navier-Stokes equations in a fixed exterior domains. We prove the existence and stability of bounded mild solutions in time t to ONSE in three-dimensional exterior domains when the coefficients are time dependent . Our method is based on the L^p-L^q-estimates of the evolution family (U(t,s)) and that of its gradient to prove boundedness of solution to linearized equations. After, we use fixed-point arguments to obtain the result on boundedness of solutions to non-linearized equations when the data belong to L^p-space and are sufficiently small. Finally, we prove existence and polynomial stability of bounded solutions to ONSE with the same condition. Our result must be useful for the study the time-periodic mild solution to the non-autonomous Oseen-Navier-Stokes equations in an exterior domains.