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

Consider the nonstationary Stokes equations in exterior domains Ω ⊂ ℝⁿ (n ≥ 3) with the compact boundary ∂Ω. We show first that the solution u(t) decays like ∥u(t)∥_r = O(t^-n/2(1-1/r)) for all 1 < r ≤ ∞ as t → ∞. This decay rate n/2(1 - 1/r) is optimal in the sense that ∥u(t)∥_r = o(t^-n/2(1-1/r)) for some 1 < r ≤ ∞ as t → ∞ occurs if and only if the net force exerted by the fluid on ∂Ω is zero.