abstract
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It is well known that the regularity of solutions to Navier-Stokes equation is controlled by the boundedness in time of the enstrophy. However, there is no proof of the existence of such bound. In fact, standard estimates for the instantaneous rate of growth of the enstrophy lead to finite time blow up, when straightforward time integration of the estimate is used. Moreover, there is recent numerical evidence to support the sharpness of these instantaneous estimates for any given instant of time. The central question is therefore, how to extend these instantaneous estimates to a finite time interval (0,T] in such a way that the dynamics imposed by the PDE are taken into account.
We state the problem of saturation of finite time estimates for enstrophy growth as an optimization problem, where the cost functional is the total change of enstrophy in a given time interval. We provide an iterative algorithm to solve the optimization problem using Viscous Burgers Equation (VBE) as a "toy" version of Navier-Stokes equation. We give numerical evidence that analytic finite time estimates for enstrophy growth in VBE are conservative, in the sense that they are not saturated.