Long-time behaviors and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type

TL;DR

This paper studies the long-term behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type in one dimension, showing convergence to explicit traveling waves and their stability.

Contribution

It provides explicit traveling wave solutions and proves their convergence and stability for a class of hyperbolic systems, including physical models.

Findings

Entropy solutions converge to traveling waves in L^1 norm over time

Explicit formulas for traveling waves based on initial data

Stability of entropy solutions in L^1 norm

Abstract

We show that in one space dimension, a linearly degenerate hyperbolic system of rich type admits exact traveling wave solutions if the initial data are Riemann type outside of a space interval. In a particular case of the system including physical models, we prove the convergence of entropy solutions to traveling waves in the $L^1$ norm as the time goes to infinity. The traveling waves are determined explicitly in terms of the initial data and the system. We also obtain the stability of entropy solutions in $L^1$.