Large-time asymptotic stability of Riemann shocks of scalar balance laws

TL;DR

This paper proves the large-time stability of Riemann shock solutions in scalar hyperbolic balance laws, showing exponential convergence under dissipative source terms and piecewise regular perturbations.

Contribution

It establishes the orbital stability and exponential convergence of Riemann shocks in scalar balance laws with dissipative sources, extending stability analysis to perturbed conditions.

Findings

Proves large-time orbital stability of Riemann shocks.

Shows exponential convergence to a shifted state.

Demonstrates stability under piecewise regular perturbations.

Abstract

We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first order scalar hyperbolic balance laws, under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover the convergence towards a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.