  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 weak solution is unique. In
this paper we prove that there exists a number u+ such that for all initial
velocities in a ball of radius u+, the Navier-Stokes equations have unique
strong global in time solutions, and that the corresponding weak solution is
unique.
