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Finite-Time Blow-Up Problem and the Maximum Growth of Palinstrophy

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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.

Author(s):

Bartosz Protas    
McMaster University
Canada

Diego Ayala    
McMaster University
Canada