DG-MP-PDE Seminar: Walter Craig (McMaster)

We consider the situation in which a weak solution of the Navier-Stokes equations fails to be continuous in the strong L^2 topology at some singular time t=T. We identify a closed set S_T in space on which the L^2 norm concentrates at this time T. The famous Caffarelli, Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of this set. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T*(R^3) above this set. Our main result is a lower bound on the L^2 concentration set. Namely, that L^2 concentration can only occur on subsets of T*(R^3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the Navier-Stokes equations which have sufficiently smooth initial data.