Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation

TL;DR

This paper analyzes the maximum growth rate of enstrophy in the fractional Burgers equation across different regimes, combining theoretical estimates with numerical maximization to understand extreme behaviors and the transition to blow-up.

Contribution

It provides sharp estimates and numerical validation for the maximum enstrophy growth rate in the fractional Burgers equation, revealing smooth transitions across regimes and detailed maximizer structures.

Findings

Maximum enstrophy growth rate estimates are sharp and match numerical maximizers.

The power-law dependence of growth rate on dissipation exponent is consistent across regimes.

Maximum growth behavior exhibits nontrivial features for small dissipation exponents.

Abstract

This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blow-up in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.