## Departmental PhD Thesis Exam – Evan Miller

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, July 16, 2019
1:10 p.m.
BA6183

PhD Candidate:  Evan Miller
Supervisor:   Robert McCann
Thesis title:  The Navier-Stokes strain equation with applications to enstrophy growth and global regularity

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The resulting identity allows us to prove a new family of scale-critical necessary and sufficient conditions for blow-up of the solution in finite time $T_{max}<+\infty$, which depend only on the history of the positive part of the second eigenvalue of the strain matrix. Since this matrix is trace-free,
this severely restricts the geometry of any finite-time blow-up.  This regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix.

We then consider a vorticity approach to the question of almost two-dimensional initial data, using this same identity for enstrophy growth and an isometry relating the third column of the strain matrix to the first two components of the vorticity. We prove a new global regularity result for initial data with two components of the vorticity sufficiently small. Finally, we prove the existence and stability of blowup for a toy model ODE of the strain equation.

A copy of the thesis can be found: Miller_Thesis_Draft