On maximum enstrophy growth in a hydrodynamic system

Enstrophy E plays an important role in the study of regularity of solutions to the three-dimensional Navier–Stokes equation. The best estimates for its growth available to-date do not rule out the possibility of enstrophy becoming unbounded in finite time which would indicate loss of regularity of solutions. It is therefore interesting to investigate sharpness of such finite-time bounds for enstrophy growth. We consider this question in the context of Burgers equation which is used as a “toy model”. The problem of saturation of finite-time estimates for the enstrophy growth is stated as a PDE-constrained optimization problem max ϕ [ E ( T ) − E ( 0 ) ] subject to E ( 0 ) = E 0 , where the control variable ϕ represents the initial condition, which is solved numerically using an adjoint-based gradient method for a wide range of time windows T and initial enstrophies E 0 . We show that this optimization problem admits a discrete family of maximizers parameterized by the wavenumber m whose members are rescaled copies of the fundamental maximizer corresponding to m = 1 . It is found that the maximum enstrophy growth in finite-time scales with the initial enstrophy as E 0 α where α ≈ 3 / 2 . The exponent is smaller than α = 3 predicted by analytic means, therefore suggesting possible lack of sharpness of analytical estimates.