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

Consider the Navier–Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity −h(t)u∞ with constant vector u∞ ∈ R3{0}. Finn raised the question whether his steady solutions are attainable as limits for t → ∞ 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∞. We study some generalized situation in which unsteady solutions start from large motions being in L3. We then conclude that the steady solutions for small u∞ 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∞ is.