Asymptotic stability theorems for viscous fluid motions in exterior domains (Q799431)

This is a very nice paper dealing with the problem of energy stability for viscous flows in an exterior domain. Since the Poincaré inequality fails for such domains the usual energy approach [see e.g. \textit{D. D. Joseph}, Stability of fluid motions, Vol. I (1976; Zbl 0345.76022)] breaks down. This writer deals rigorously with both steady and unsteady base flows and gives an elegant proof that if the Reynolds number is less than a suitably defined energy stability limit the (nonlinear) perturbed motion decays in the sense \(\sup_{\Omega}|\underset \tilde{} u(\underset \tilde{} x,t)| =O(t^{-1/2}),\) where \(\Omega\) is the exterior region occupied by the fluid and \(\underset \tilde{} u\) is the velocity. Precise estimates are also given for the decay rate of \(\underset \tilde{} u_ t\) and \(\nabla\underset \tilde{} u\).