  In this paper, we simplify and extend the results of \cite{GZ} to include the
case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space
of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb
H}[{\mathbb{R}}^3]^3 =: {\mathbb H}$ be the completion of the set, ${{{\bf{u}}
\in (\mathbb {C}_0^\infty [ \R^3 ])^3. {} | \nabla \cdot {\bf{u}} = 0}}$, with
respect to the inner product of ${[L^2({\mathbb{R}}^3)]^3} $. In this paper, we
consider sufficiency conditions on a class of functions in ${\mathbb H}$ which
allow global-in-time strong solutions to the three-dimensional Navier-Stokes
equations on ${\mathbb {R}}^3$. These equations describe the time evolution of
the fluid velocity and pressure of an incompressible viscous homogeneous
Newtonian fluid in terms of a given initial velocity and given external body
forces. Our approach uses the analytic nature of the Stokes semigroup to
construct an equivalent norm for $\mathbb{H}$ which allows us to prove a
reverse of the Poincar\'e inequality. This result allows us to provide strong
bounds on the nonlinear term. We then prove that, under appropriate conditions,
there exists a positive constant $ {{u}_+}$, depending only on the domain, the
viscosity and the body forces such that, for all functions in a dense set
$\mathbb{D}$ contained in the closed ball ${{\mathbb B} ({\mathbb {R}}^3)}=:
{\mathbb B}$ of radius $ (1/2){{u}_ +} $ in ${\mathbb {H}}$, the Navier-Stokes
equations have unique strong solutions in ${\mathbb C}^{1}
({(0,\infty),{\mathbb {H}}})$.
