Navier-Stokes Flow Past a Rigid Body: Attainability of Steady Solutions as Limits of Unsteady Weak Solutions, Starting and Landing Cases

摘要

Consider the Navier-stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity - h(t)u(infinity) with constant vector U-infinity is an element of R-3 backslash{0}. Finn raised the question whether his steady solutions are attainable as limits for t - infinity 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 et al. (Arch Ration Mech Anal 138:307-318, 1997) for small u(infinity). 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-infinity 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-infinity is.