Slow motion of internal shock layers for the Jin-Xin system in one space dimension

TL;DR

This paper investigates the slow, metastable movement of shock layers in the Jin-Xin system, revealing that the internal transition layer drifts toward equilibrium at an exponentially slow rate, supported by rigorous analysis and numerical validation.

Contribution

It provides a rigorous and asymptotic analysis of the metastable shock layer dynamics in the Jin-Xin system, deriving an ODE for the layer position and confirming it through numerical computations.

Findings

The shock layer drifts exponentially slowly towards equilibrium.

An ODE describing the layer's position is derived and validated.

The analysis combines rigorous and asymptotic methods.

Abstract

This paper considers the slow motion of the shock layer exhibited by the solution to the initial-boundary value problem for a scalar hyperbolic system with relaxation. Such behavior, known as metastable dynamics, is related to the presence of a first small eigenvalue for the linearized operator around an equilibrium state; as a consequence, the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval. In this contest, both rigorous and asymptotic approaches are used to analyze such slow motion for the Jin-Xin system. To describe this dynamics we derive an ODE for the the position of the internal transition layer, proving how it drifts towards the equilibrium location with a speed rate that is exponentially slow. These analytical results are also validated by numerical computations.