Abstract
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.
Highlights
This paper concerns weak solution of the Navier – Stokes equations in three dimensions
The energy inequality implies that u(·, t) L2 is in L∞ t [0, ∞) as a function of time, and it is well known that solutions are continuous in the weak L2 topology
At a time T of L2 discontinuity we identify the L2 concentration set STL2, which is a closed set in space, as a subset of the singular set at that time T, namely STL2 ⊆ ST (u) := S(u) ∩ {t = T }
Summary
This paper concerns weak solution of the Navier – Stokes equations in three dimensions. The energy inequality implies that u(·, t) L2 is in L∞ t [0, ∞) as a function of time, and it is well known that solutions are continuous in the weak L2 topology. The size of the L2 concentration set is bounded above by the theory of partial regularity, and in particular a corollary of the results in the well known paper of Caffarelli, Kohn & Nirenberg [5] implies that for any singular time T > 0 the size of the singular set (in space) of the solution is bounded above.
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