On the Size of the Navier - Stokes Singular Set

A beautiful and influential subject in the study of the question of smoothness of solutions for the Navier – Stokes equations in three dimensions is the theory of partial regularity. A major paper on this topic is Caffarelli, Kohn & Nirenberg [5](1982) which gives an upper bound on the size of the singular set S(u) of a suitable weak solution u. In the present paper we describe a complementary lower bound. More precisely, we study the situation in which a weak solution fails to be continuous in the strong L topology at some singular time t = T . We identify a closed set in space on which the L norm concentrates at this time T , and we study microlocal properties of the Fourier transform of the solution in the cotangent bundle T (R) above this set. Our main result is that L concentration can only occur on subsets of T (R) 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.