Persistence of generalized roll-waves under viscous perturbation

TL;DR

This paper investigates how periodic solutions with shocks in hyperbolic systems persist when small viscous effects are introduced, using expansions and Green's function estimates.

Contribution

It provides a rigorous proof of the persistence of periodic roll-wave solutions under small viscous perturbations.

Findings

Periodic roll-wave solutions persist under viscous perturbation.

Expansion techniques effectively analyze viscous effects on hyperbolic solutions.

Green's function estimates are crucial for the proof.

Abstract

The purpose of this article is to study the persistence of solution of a hyperbolic system under small viscous perturbation. Here, the solution of the hyperbolic system is supposed to be periodic: it is a periodic perturbation of a roll-wave. So, it has an infinity of shocks. The proof of the persistence is based on an expansion of the viscous solution and estimates on Green's functions.