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