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

Abstract

Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $Ω$ 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 $\partialΩ$, friction coefficient $α(x)$ and the outflow $u_\infty$. 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 $α(x)+u_\infty\cdotν(x)/2\geq 0$ for every $x\in\partialΩ$, where $ν(x)$ stands for the outward unit normal to the boundary $\partialΩ$.