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

TL;DR

This paper investigates the decay behavior of the Oseen flow in exterior domains with Navier slip-with-friction boundary conditions, establishing decay estimates and analyzing the resolvent's regularity near zero.

Contribution

It provides new decay estimates for the Oseen semigroup in exterior domains under slip-with-friction conditions, linking boundary geometry, friction, and outflow effects.

Findings

Established resolvent set conditions based on boundary and flow parameters.

Derived $L^q$-$L^r$ decay estimates for the Oseen semigroup.

Analyzed the resolvent's regularity near the origin in the complex plane.

Abstract

Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $\Omega$ 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 $\partial\Omega$, friction coefficient $\alpha(x)$ 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 $\alpha(x)+u_\infty\cdot\nu(x)/2\geq 0$ for every $x\in\partial\Omega$, where $\nu(x)$ stands for the outward unit normal to the boundary $\partial\Omega$.