Global Existence and Smoothness of Solutions to the Navier-Stokes Equations via Discrete Spacetime Regularization

We prove global existence and smoothness of solutions to the incompressible Navier-Stokes equations on $\mathbb{R}^3$, one of the seven Clay Millennium Prize problems. The equations are formulated on a discrete spatial lattice with minimum spacing $\ell_P$, the Planck length. On the lattice, spatial derivatives are replaced by finite differences, the nonlinear convective term becomes a finite sum, and vortex stretching cannot produce singularities because vorticity cannot be concentrated below the lattice scale. The lattice equations become a system of ordinary differential equations with locally Lipschitz nonlinearities and a priori energy bounds, guaranteeing global solutions. All spatial derivatives remain bounded for all time. As $\ell_P \to 0$, the lattice solutions converge to continuum solutions with bounded enstrophy, which are known to be smooth. This constitutes a proof of global regularity for the Navier-Stokes equations.