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 $L^2$ topology at some singular time $t=T$. We identify a closed set in space on which the $L^2$ norm concentrates at this time $T$, and 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 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.