From Jean Leray to the millennium problem: the Navier–Stokes equations

One of the Millennium problems of Clay Mathematics Institute from 2000 concerns the regularity of solutions to the instationary Navier–Stokes equations in three dimensions. For an official problem description, we refer to an article by Charles Fefferman (Existence and smoothness of the Navier–Stokes equation. http://www.claymath.org/sites/default/files/navierstokes.pdf) on the whole space problem as well as the periodic setting in ℝ³. In this article, we will introduce the Navier–Stokes equations, describe their main mathematical problems, discuss several of the most important results, starting from 1934 with the seminal work by Jean Leray, and proceeding to very recent results on non-uniqueness and examples involving singularities. On this tour de force we will explain the open problem of regularity.