On the Maximum Enstrophy Growth in Burgers Equation

The regularity of solutions of the three-dimensional Navier-Stokes equation is controlled by the boundedness of the enstrophy ?. The best estimate available to-date for its rate of growth is d?/dt ? C?3, where C > 0, which was recently found to be sharp by Lu & Doering (2008). Applying straightforward time-integration to this instantaneous estimate leads to the possibility of loss of regularity in finite time, the so-called blow-up, and therefore the central question is to establish sharpness of such finite-time bounds. We consider an analogous problem for Burgers equation which is used as a toy model. The problem of saturation of finite-time estimates for the enstrophy growth is stated as a PDE-constrained optimization problem where the control variable represents the initial condition, which is solved numerically for a wide range of time windows T > 0 and initial enstrophies ?0. We find that the maximum enstrophy growth in finite time scales as ?0? with ? ? 3/2. The exponent is smaller than ? = 3 predicted by analytic means, therefore suggesting lack of sharpness of analytical estimates.