Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations

TL;DR

This paper analyzes the long-term behavior of traveling periodic waves in viscous conservation laws, establishing stability under various perturbations and linking it to Whitham modulation systems.

Contribution

It provides a unified theoretical framework for stability analysis and classifies systems based on phase-coupling, including new results on nonlinear asymptotic stability.

Findings

Long-time behavior governed by second-order Whitham systems

Classification of systems into phase-coupling and decoupling cases

Nonlinear asymptotic stability for localized phase-decoupled perturbations

Abstract

We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated second-order formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate to initial data for this formal system, a task accomplished by detailed estimates on the linearized solution operator about the background wave. Notably, our approach gives both a common theoretical treatment and a complete classification in terms of "phase-coupling" or "-decoupling" of general systems of conservation or balance laws, encompassing cases that had previously been studied separately or not at all. At the same time, our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase-decoupled case, further distinguishing behavior in the different cases. An interesting technical aspect of our analysis is that for systems of conservation laws the Whitham modulation description is of system rather than scalar form, as a consequence of which renormalization methods such as have been used to treat the reaction-diffusion case in general do not seem to apply.