Large time decay of the Oseen flow in exterior domains subject to the Navier slip-with-friction boundary condition

Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $\mathrm{\Omega }$ subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary $\mathrm{\partial }\mathrm{\Omega }$, friction coefficient $\alpha (x)$ and the outflow $u_{\mathrm{\infty }}$. We then study the regularity of the resolvent near the origin in the complex plane to develop $L^q$-$L^r$ decay estimates of the Oseen semigroup provided that $\alpha (x)+u_{\mathrm{\infty }}\cdot \nu (x)\mathrm{/}2\ge 0$ for every $x\in \mathrm{\partial }\mathrm{\Omega }$, where $\nu (x)$ stands for the outward unit normal to the boundary $\mathrm{\partial }\mathrm{\Omega }$.