Large data existence of global-in-time strong solutions to the incompressible Navier-Stokes equations in high space dimensions

Abstract

We study the existence of a strong solution to the initial value problem for the incompressible Navier-Stokes equations in the whole space. Our investigation shows that a ``suitable'' weak solution to the problem becomes a strong one whenever the initial velocity is divergence free and uniformly bounded with finite energy. Our results seem to have given a positive answer to the Navier-Stokes millennium problem proposed by the Clay Mathematical Institute.