  In this paper we take a new approach to a proof of existence and uniqueness
of solutions for the 3D-Navier-Stokes equations, which leads to essentially the
same proof for both bounded and unbounded domains and for homogeneous or
inhomogeneous incompressible fluids. Our approach is to construct the largest
separable Hilbert space ${\bf{SD}}^2[\R^3]$, for which the Leray-Hopf (type)
solutions in $L^2[{\mathbb R}^3]$ are strong solutions in ${\bf{SD}}^2[\R^3]$.
We say Leray-Hopf type because our solutions are weak in the spatial sense but
not in time.
