  A well-known unsolved problem (in the classical theory of fluid mechanics) is
to identify a set of initial velocities, which may depend on the viscosity, the
body forces and possibly the boundary of the fluid that will allow global in
time solutions to the three-dimensional Navier-Stokes equations. (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.) A related problem is to provide
conditions under which we can be assured that the numerical approximation of
these equations, used in a variety of fields from weather prediction to
submarine design, have only one solution. In earlier papers, we solved this
problem for a bounded domain. In this paper, we use an approach based on
additional physical insight, that allows us to prove that there exists unique
global in time solutions to the Navier-Stokes equations on R^3.
