Asymptotic properties of the Stokes flow in an exterior domain with slowly decaying initial data and its application to the Navier-Stokes equations

In this paper, we study the decay rate of the Stokes flow in an exterior domain with a slowly decaying initial data ${\mathbf{u}}_0(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^{-\text{al}}),0<\text{al}\le n$. %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.