Global-in-Time solutions of the Navier-Stokes equations

Abstract

We establish the existence of a uniformly bounded $ C^\infty $ solution of the Navier-Stokes equations on $\mathbb{R}^3 x\ [0, \infty) $ without external forces or boundaries for a divergence free initial condition $ u_o \in \cap_m H^m $ when the viscosity $ \nu $ is $ > 0 $. Such $ C^\infty $ solution of the Navier-Stokes equations are a solution of Option A of the Millenium Prize Problem of the Clay Institute for the Navier-Stokes equations.