# Navier-Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases

## TL;DR

This paper investigates whether steady solutions of the Navier-Stokes equations in exterior domains can be obtained as limits of unsteady flows, considering both starting and landing scenarios, and extends previous results to larger initial motions.

## Contribution

It generalizes prior work by showing steady solutions are attainable from large initial motions in $L^3$, and discusses the landing case where the rest state is always reachable.

## Key findings

- Steady solutions are attainable from large initial motions in $L^3$.
- The rest state is always attainable in the landing scenario regardless of initial velocity.
- Extension of previous small velocity results to larger initial motions.

## Abstract

Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the question whether his steady slutions are attainable as limits for $t\to\infty$ of unsteady solutions starting from motionless state when $h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We study some generalized situation in which unsteady solutions start from large motions being in $L^3$. We then conclude that the steady solutions for small $u_\infty$ are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions. The opposite situation, in which $h(t)=0$ after some finite time and $h(0)=1$ (landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large $u_\infty$ is.