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Sufficiency Class for Global (in Time) Solutions to the 3D-Navier-Stokes Equations

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.