Extreme Vortex States and the Growth of Palinstrophy in Two Dimensions

We probe the sharpness of analytic estimates for the instantaneous rate of growth and the finite-time growth of palinstrophy in 2D viscous incompressible flows on periodic domains. This effort is part of a broader research program concerning a systematic search for extreme vortex states which is intrinsically related to the finite-time ``blow-up'' problem in 3D incompressible flows. Evidence is presented for the existence of a family of 2D vorticity fields parametrized by their energy and palinstrophy which saturate an estimate characterizing the finite-time growth of palinstrophy. The family of such ``optimal'' vortex states is obtained by solving suitable optimization problems in which the rate of growth of palinstrophy is maximized under constraints. Although found as a solution of an instantaneous problem, vortex states from this family also saturate the finite-time estimates. This intriguing finding leads to some open questions about the 3D case, namely, whether extreme vortex states with prescribed energy and enstrophy may exhibit a larger growth of enstrophy than the previously found fields in which only enstrophy was fixed and whose growth of enstrophy was too weak to produce a singularity in finite time.