Sharp bounds on enstrophy growth in the viscous Burgers equation

Details Sharp bounds on enstrophy growth in the viscous Burgers equation Sharp bounds on enstrophy growth in the viscous Burgers equation

2012-04-17

arxiv.org/abs/1204.3905v1

Mathematics

Analysis of PDEs

12 pages

Scientific paper

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 $\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as $\mathcal{E}^{-1/2}$. These bounds are sharp for sufficiently smooth initial conditions.

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