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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Walter A Craig
McMaster University
Title: On the size of the Navier - Stokes singular set

Abstract: Consider the hypothetical situation in which a solution $u(t,x)$ of the Navier-Stokes equations in three dimensions develops a singularity at some singular time $t = T$. It could do this by a failure of smoothness, or more seriously, the time evolution could also fail to be continuous in the strong $L^2$ topology. The famous Caffarelli Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of the singular set $S(T)$ on which smoothness fails. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle $T^\ast(R^3)$ above this set $S(T)$. Our first result is that, if $S(T)$ is nonempty, then it cannot be too small, in the sense that there is a lower bound on the size of the wave front set $WF(u(T,.))$. Namely, singularities can only occur on subsets of $T^\ast(R^3)$ which are relatively large. Furthermore, if the solution evolution is discontinuous in $L^2$ we identify a closed subset $S'(T)$ of $ S(T)$ in terms of H-measures on which the $L^2$ norm concentrates at this time $T$. We then give a lower bound on the microlocal manifestation of this $L^2$ concentration set, which is larger than the general one above. An element of the proof of these two bounds is a global estimate on weak solutions of the Navier-Stokes equations which have sufficiently smooth initial data.

Pdf File: WalterCraig.pdf
Date: Thursday, February 16, 2017
Time: 1:30 am
Location: Wean Hall 7218
Note: 1