Navier-Stokes flow past a translating body: attainability of the steady state as limit of an $L^p$ weak solution

Abstract

Consider a viscous, incompressible, Newtonian fluid that fills the region of space exterior to a rigid body. Let the fluid be activated by a perturbation $u_0\in L^p(\OO)$, $p\in (2,3)$ and let the body start to translate with a prescribed velocity $-h(t)u_\infty$, $t>0$, $u_\infty\in\R^3\setminus \{0\}$. Suppose that the body reaches the stationary regime after some finite time. We show that, under suitable assumptions on $u_\infty$, a steady state is always reachable by the fluid, asymptotically in time, as limit of a suitable weak solution to the Navier-Stokes flow in suitable metrics. Our result seems to be a further contribution related to the {\it Starting Problem}, posed by R. Finn in 1965.