Extreme vortex states and the hydrodynamic blow-up problem

2014/05/13 Tue 16:30 - 18:00
Bartosz Protas
McMaster Univ.

In the presentation we will discuss our research program
concerning the study of extreme vortex events in viscous
incompressible flows. These vortex states arise as the flows
saturating certain fundamental mathematical estimates, such as the
bounds on the maximum enstrophy growth in 3D. They are therefore
intimately related to the question of spontaneous singularity
formation in the 3D Navier-Stokes system, known as the hydrodynamic
“blow-up” problem. We demonstrate how new insights concerning such
problems can be obtained by formulating them as variational PDE
optimization problems which can be solved computationally using
suitable discrete gradient flows. In offering a systematic approach to
finding flow solutions which may saturate known estimates, the
proposed paradigm provides a bridge between mathematical analysis and
scientific computation. In particular, it allows one to determine
whether or not certain mathematical estimates are “sharp”, or if they
may still be improved. In the presentation we will review a number of
new results concerning 2D and 3D vortex flows characterized by the
maximum possible growth of, respectively, palinstrophy and
enstrophy. We will also discuss their relation to the available
theoretical bounds obtained with rigorous methods of mathematical

[Joint work with Diego Ayala]