Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Abstract

Consider the nonstationary Stokes equations in exterior domains \(\Omega \subset{\Bbb R}^n(n\ge 3)\) with the compact boundary \(\partial \Omega\). We show first that the solution \(u(t)\) decays like \(\|u(t)\|_r = O(t^{-\frac{n}{2}(1-\frac{1}{r})})\) for all \(1 < r \le \infty\) as \(t\to \infty\). This decay rate \(\frac{n}{2}(1-\frac{1}{r})\) is optimal in the sense that \(\|u(t)\|_r =o(t^{-\frac{n}{2}(1-\frac{1}{r})})\) for some \(1 < r \le \infty\) as \(t\to \infty\) occurs if and only if the net force exerted by the fluid on \(\partial\Omega\) is zero.