  In this paper we focus on the Cauchy problem for the incompressible
Navier-Stokes equation with a rough external force. If the given rough external
force is small, we prove the local-in-time existence of this system for any
initial data belonging to the critical Besov space
$\dot{B}_{p,p}^{-1+\frac{3}{p}}$, where $3<p<\infty$. Moreover, We show the
long-time behavior of the priori global solutins constructed by us. Also, we
give three kinds of uniqueness results of the forced Navier-Stokes equations.
