Maximum Enstrophy Dissipation in 2D Navier-Stokes Flows in the Limit of Vanishing Viscosity

We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in the limit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This problem is solved numerically and the dependence of the maximum enstrophy dissipation on viscosity is shown to be in quantitative agreement with the estimate due to Ciampa, Crippa & Spirito (2021), demonstrating the sharpness of this bound. Furthermore, to probe this and related problems from the kinetic theory perspective, we also introduce a velocity-discretized Boltzmann equation with a simplified BGK collision operator that lead to the incompressible Navier-Stokes equations in the hydrodynamic limit. Moreover, we highlight the utilization of numerical computations conducted in 2D to provide information about the rate with which this hydrodynamic limit is achieved when the Knudsen number tends to zero.