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 -µ to 0 and satisfy the Navier-Stokes system with no right hand side.
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.
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.
part of the course will be addressed ancient solutions in a half space in
connection with possible blow-ups at the boundary.