An Explanation of Metastability in the Viscous Burgers Equation with Periodic Boundary Conditions via a Spectral Analysis

Abstract

A “metastable solution to a differential equation typically refers to a family of solutions for which solutions with initial data near the family converge 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.