An explanation of metastability in the viscous Burgers equation with periodic boundary conditions via a spectral analysis

TL;DR

This paper investigates the metastable behavior in the viscous Burgers equation with periodic boundary conditions by spectral analysis, aiming to understand fluid flow dynamics without relying on classical transformations.

Contribution

It introduces a new spectral analysis approach to explain metastability in the Burgers equation, potentially applicable to more complex fluid systems like Navier-Stokes.

Findings

Identification of a candidate metastable family

Spectral analysis explains metastable behavior

Method may extend to realistic fluid models

Abstract

A "metastable solution" to a differential equation typically refers to a family of solutions for which nearby initial data converges to the family much faster than evolution along the family. Metastable families have been observed both experimentally and numerically in various contexts, they are believed to be particularly relevant for organizing the dynamics of fluid flows. In this work we propose a candidate metastable family for the Burgers equation with periodic boundary conditions. Our choice of family is motivated by our numerical experiments. We furthermore explain the metastable behavior of the family without reference to the Cole--Hopf transformation, but rather by linearizing the Burgers equation about the family and analyzing the spectrum of the resulting operator. We hope this may make the analysis more readily transferable to more realistic systems like the Navier--Stokes equations. Our analysis is motivated by ideas from singular perturbation theory and Melnikov theory.