Sharp bounds on enstrophy growth in the viscous Burgers equation

TL;DR

This paper establishes sharp bounds on the maximum enstrophy growth and the time to reach it in the viscous Burgers equation, using analytical methods to justify numerical findings.

Contribution

It provides rigorous analytical bounds on enstrophy growth in the viscous Burgers equation, confirming numerical results and characterizing the scaling behavior.

Findings

Maximum enstrophy scales as E^{3/2}

Time to reach maximum scales as E^{-1/2}

Bounds are sharp for smooth initial conditions

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 $\mathcal{E}^{3/2}$, 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.