Sharp bounds on enstrophy growth for viscous scalar conservation laws

TL;DR

This paper establishes precise bounds on how rapidly enstrophy can grow in viscous scalar conservation laws, confirming a conjecture for the viscous Burgers equation through rigorous analysis.

Contribution

It provides sharp, theoretically proven bounds on enstrophy growth, linking it to viscous shock solutions and resolving a conjecture from prior numerical studies.

Findings

Upper bounds on enstrophy growth are tight and match the enstrophy produced by steepest viscous shocks.

The bounds are valid under $L^ Infty$ and total variation constraints and depend on viscosity.

The results confirm the conjectured behavior of enstrophy growth in viscous Burgers equation.

Abstract

We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the $L^\infty$ and total variation bounds and viscosity. This answers a conjecture by D. Ayala and B. Protas (\emph{Physica D}, 2011), based on numerical evidence, for the viscous Burgers equation.