# Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle

## TL;DR

This paper studies the long-term behavior of a generalized Oseen evolution operator and applies the findings to analyze the flow of viscous incompressible fluids past a rotating obstacle, extending understanding of fluid dynamics in exterior domains.

## Contribution

It develops $L^q$-$L^r$ decay estimates for the evolution operator and applies these to the Navier-Stokes equations in exterior domains with moving obstacles.

## Key findings

- Established $L^q$-$L^r$ decay estimates for the evolution operator.
- Applied decay estimates to the Navier-Stokes initial value problem.
- Extended previous linear analysis to nonlinear fluid flow scenarios.

## Abstract

Consider the motion of a viscous incompressible fluid in a 3D exterior domain when a rigid body moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $L^q$-$L^r$ smoothing action near the initial time as well as generation of the evolution operator was shown by Hansel and Rhandi (J. Reine Angew. Math. 2014) under reasonable conditions. In this paper we develop the $L^q$-$L^r$ decay estimates of the evolution operator and then apply them to the Navier-Stokes initial value problem.