Navier-Stokes Equations: The Beauty and the Beast

As is well-known, the Navier-Stokes equations are at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Oil Industry, Airplane, Ship and Car Industries, etc. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. However, in spite of these successes, to date, a number of unresolved basic questions — mostly, for the physically relevant case of three-dimensional (3D) motions — remain open. Among them, certainly, the most famous is that of proving or disproving existence of 3D regular solutions for all times and for data of arbitrary ‘size’, no matter how smooth. This notorious question has challenged several generations of mathematicians since the beginning of the 20th century who, yet, have not been able two furnish a complete answer. The problem has become so obsessing and intriguing that, as is known, mathematicians have decided to put a generous bounty on it. In fact, properly formulated, it is listed as the third of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. It should be observed that the analogous question in the two-dimensional (2D) case received a positive answer about half a century ago. In this talk I shall present the main known results of existence, uniqueness and regularity of solutions to the corresponding initial-boundary value problem in a way that should be accessible also to non-specialists. Moreover, I will furnish a number of significant open questions and explain why the current mathematical approaches fail to answer them. In some cases, I shall also point out possible strategies of resolution.