Enstrophy growth in the viscous Burgers equation

TL;DR

This paper investigates the bounds and asymptotic behavior of enstrophy growth in solutions to the viscous Burgers equation, revealing sharp bounds, shock formation dynamics, and scaling laws for maximum enstrophy on different domains.

Contribution

It provides a variational characterization of enstrophy growth maximizers, constructs exact self-similar solutions, and establishes scaling laws for maximum enstrophy and the time to reach it.

Findings

Maximizer of enstrophy growth is sharp but does not saturate Poincaré inequality.

Asymptotic representation of maximizers as viscous shocks on rarefactive waves.

Maximum enstrophy scales as E^{3/2} with initial enstrophy E, with logarithmic correction for time to reach maximum.

Abstract

We study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy's rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the Poincar\'e inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to prove that a larger growth of enstrophy can be achieved when the initial data to the viscous Burgers equation saturates the Poincar\'e inequality up to a numerical constant. An exact self-similar solution of the Burgers equation is constructed to describe formation of a metastable viscous shock on the background of a linear rarefactive wave. When we consider the Burgers equation on an infinite line subject to the nonzero (shock-type) boundary conditions, we prove 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 $\mathcal{E}^{-1/2} \log(\mathcal{E})$. Similar but slower rates are proved on the unit circle.