Finite-time blow-up problem and the maximum growth of palinstrophy

Copyright © ETC 2013 - 14th European Turbulence Conference.All rights reserved. This investigation is a part of a broader research effort aiming to discover solutions of the Navier-Stokes system in 2D and 3D which can saturate certain analytically obtained bounds on the maximum growth of enstrophy and palinstrophy [4, 5]. This research is motivated by questions concerning the possibility of finite-time blow-up of solutions of the 3D Navier-Stokes system where such estimates play a key role. We argue that insights concerning the sharpness of these estimates can be obtained from the numerical solution of suitably-defined PDE optimization problems. In the present contribution we focus on the sharpness of the analytical bounds on the instantaneous rate of growth of palinstrophy P in 2D incompressible flows and identify the vortex configurations which maximize this quantity under certain constraints. These optimal vortex states exhibit a distinct scale-invariant structure and offer insights about mechanisms leading to generation of small scales in the enstrophy cascade.