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

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.