Cauchy problem for the incompressible Navier-Stokes equation with an external force

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 $3<p<\mathrm{\infty }$. 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.