July  2010, 28(3): 1165-1178. doi: 10.3934/dcds.2010.28.1165

On the size of the Navier - Stokes singular set

1. 

International Institute of the Earthquakes Prediction, Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow 117997, Russian Federation

2. 

Department of Mathematics & Statistics, McMaster University, Hamilton Ontario L8S 4K1, Canada

Received  April 2010 Published  April 2010

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.
Citation: Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165
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