Using global invariant manifolds to understand metastability in Burgers equation with small viscosity

TL;DR

This paper uses invariant manifolds to explain the long-term metastable behavior of solutions to Burgers equation with small viscosity, showing how solutions linger near diffusive N-waves before settling into a stable diffusion wave.

Contribution

It provides a geometric and invariant manifold framework to understand metastability in Burgers equation, linking solution dynamics to global center and invariant manifolds.

Findings

Solutions exhibit metastability by lingering near diffusive N-waves

Invariant manifolds characterize the long-time behavior of solutions

Solutions eventually converge to a self-similar diffusion wave

Abstract

The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this "metastable" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.