Limits on Enstrophy Growth for Solutions of the Three-dimensional Navier-Stokes Equations

The enstrophy, the square of the L-2 norm of the vorticiry field, is a key quantity for the determination of regularity and uniqueness properties for solutions to the Navier-Stokes equations. In this paper we investigate the maximal enstrophy generation rate for velocity fields with a fixed amount of enstrophy, as a function of the magnitude of the enstrophy via numerical solution of the Euler-Lagrange equations for the associated variational problem. The veracity of the novel computational scheme is established by utilizing the exactly soluble version of the problem for Burgers' equation as a benchmark. The results for the three dimensional Navier-Stokes equations are found to saturate functional estimates for the maximal enstrophy production rate as a function of enstrophy.