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

On the size of the Navier - Stokes singular set

Maxim Arnold 1, and  Walter Craig 2,

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.
Keywords: regularity theory, fluid dynamics..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165
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