Maximum Enstrophy Growth in Burgers Equation

The regularity of solutions of the Navier--Stokes equation is controlled by the boundedness of the enstrophy $\mathcal{E}$. The best estimate for its rate of growth is $d\mathcal{E}/dt \leq C\mathcal{E}^{3}$, for $C>0$, leading to the possibility of a finite--time blow--up when straightforward time integration is used. Recent numerical evidence by Lu \& Doering (2008) supports the sharpness of the instantaneous estimate. Thus, the central question is how to extend the instantaneous estimate to a finite--time estimate in a way that will incorporate the dynamics imposed by the PDE. We state the problem of saturation of finite--time estimates for the enstrophy growth as a PDE--constrained optimization problem, using the Burgers equation as a ``toy model''. The following problem is solved numerically: \begin{displaymath} \max_{\phi}[\mathcal{E}(T) - \mathcal{E}(0)]\quad\mbox{subject to}\quad\mathcal{E}(0) = \mathcal{E}_0 \end{displaymath} where $\phi$ represents the initial data for Burgers equation, for a wide range of values of $T>0$ and $\mathcal{E}_0$ finding that the maximum enstrophy growth in finite time scales as $\mathcal{E}^{\alpha}_0$ with $\alpha\approx 3/2$, an exponent different from $\alpha = 3$ obtained by analytic means.