Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium 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.Recording: http://mm.math.cmu.edu/recordings/cna/walter_craig_small.mp4Pdf File: WalterCraig.pdfDate: Thursday, February 16, 2017Time: 1:30 amLocation: Wean Hall 7218Note: 1