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.