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.

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