The Navier-Stokes regularity problem.

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier-Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted. This article is part of the theme issue 'Stokes at 200 (Part 1)'.