  Abstract. Using Constantin-Iyer representation also known more generally as
Euler-Lagrangian approach, we prove that there exists a unique local solution
in weighted Sobolev spaces of the following Navier-Stokes equations with
forcing on the full space $\mathbf{R}^{d}$, for any dimension d and p such that
$p>d\geq2$ and $\epsilon>0 $,
  $\displaylines{ d\mathbf{u}\left(t\right) =
\left[\mathcal{S}\left(-u^{k}\left(t\right)\partial_{k}\mathbf{u}\left(t\right)\right)+\frac{\epsilon^{2}}{2}\Delta\mathbf{u}\left(t\right)+G\left(t\right)\right]dt,
\mathbf{u}\left(0\right) = \mathbf{u}_{0},\hspace{1em}\text{div
}\mathbf{u}=0.}$
