Concentration vs regularity in the Navier-Stokes equations

The three-dimensional Navier-Stokes equations describe the motion of incompressible viscous fluids. They date back to the 19th century. A breakthrough in their mathematical analysis came from the pioneering work of Leray in 1934. As far as we know, many questions about the behavior of solutions remain open, notably the uniqueness of weak solutions and their regularity or finite time blow-up.

In this talk, we will survey some aspects of the regularity theory for the Navier-Stokes equations. The other side of the coin is finding necessary conditions for solutions developing finite time singularities. In a recent work with Y. Maekawa (Kyoto University) and H. Miura (Tokyo Tech) we found a concentration phenomenon for blowing-up solutions. We will explain this result and a strengthened version, which is work in progress with T. Barker (ENS Paris).