Extreme Vortex States and Singularity Formation in Incompressible Flows

Abstract

One of the most prominent open problems in mathematical physics is determining whether solutions to the incompressible three-dimensional (3D) Navier-Stokes system, corresponding to arbitrarily large smooth initial data, remain regular for arbitrarily long times. A promising approach to this problem relies on the fact that both the smoothness of classical solutions and the uniqueness of weak solutions in 3D flows are ultimately controlled by the growth properties of the $H^1$ seminorm of the velocity field U, also known as the enstrophy.

In this context, the sharpness of analytic estimates for the instantaneous rate of growth of the $H^2$ seminorm of U in two-dimensional (2D) flows, also known as palinstrophy, and for the instantaneous rate of growth of enstrophy in 3D flows, is assessed 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.