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.

Cited by in corpus (2)

• Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains
• Existence of a Stationary Navier-Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions