  In this paper, we study the decay rate of the Stokes flow in an exterior
domain with a slowly decaying initial data ${\bf u}_0(x)=O(|x|^{-\al}),
0<\al\leq n$. %which is not $L^1$ integrable. As an application we find the
unique strong solution of the Navier-Stokes equations corresponding to a slowly
decaying initial data. We also derive the pointwise decay estimate of the
Navier-Stokes flow. Our decay rates will be optimal compared with the decay
rates of the heat flow.
