Ancient solutions to the Navier-Stokes equations
Gregory Seregin (Oxford)

The main aim of the course is to explain how the so-called ancient (backward) solutions appear in the theory of regularity for the Navier-Stokes equations. Those solutions are defined in the whole space and in time from - to 0 and satisfy the Navier-Stokes system with no right hand side.

The existence of non-trivial ancient solutions can be regarded as a necessary condition for possible blow-ups in the Navier-Stokes theory. We are going to discuss in details a special subclass of ancient solutions called mild bounded ancient solutions. A conjecture on them will be stated. It has a form of Liouville type theorem. The validity of this conjecture would rule out blow-ups of type I.

In the first part of course, the notion of ancient solutions will be introduced and certain properties of mild bounded ancient solutions will be demonstrated.

A procedure, relating possible blow-ups to non-trivial mild bounded ancient solutions, will be discussed in the second part of the course.

In the third part of the course, we shall consider several cases for which the above mentioned conjecture is valid and explain why it is so.

Our fourth part of the course will be addressed ancient solutions in a half space in connection with possible blow-ups at the boundary.