Sharp bounds on enstrophy growth in the viscous Burgers equation

Abstract

We use the Cole-Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle.We show that the maximum enstrophy achieved in the time evolution is scaled as E3/2, where E is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as E?1/2. These bounds are sharp for initial conditions given by odd C3 functions that are convex on half-period. © 2012 The Royal Society.