Optimal vortex states and singularity formation in incompressible flows

Diego Ayala
University of Michigan
Mathematics and Statistics

We investigate the sharpness of analytic estimates for the instantaneous rate of growth of
palinstrophy in 2D flows and for the instantaneous rate of growth of enstrophy in 3D flows, by numerically solving suitable constrained optimization problems. It is found that the instantaneous estimates for both 2D and 3D flows are saturated by highly localized vortex structures.

Moreover, finite-time estimates for the total growth of palinstrophy in 2D and enstrophy
in 3D are obtained from the corresponding instantaneous estimates and, by using the
(instantaneously) optimal vortex structures as initial conditions in the Navier-Stokes system and numerically computing their time evolution, the finite-time estimates are found to be uniformly sharp for 2D flows, and sharp over increasingly short time intervals for 3D flows. Although computational in essence, these results indicate a possible route for finding an extreme initial condition for the Navier-Stokes system that could lead to the formation of a singularity in finite time.

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