Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness

This paper presents the main results concerning solubility of the basic initial-boundary value problem and the Cauchy problem for the three-dimensional non-stationary Navier–Stokes equations, together with a list of what to prove in order to solve the sixth problem of the “seven problems of the millennium” proposed on the Internet at the site http://claymath.org/.