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 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.

Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation

“…Hence, differently from the cases considered in [22,23,31], here the metastable behavior is characterized by the fact that the steady state U − ε,1 is unstable, so that the solutions starting from an initial configuration close to U − ε,1 are pushed away towards their asymptotic limit, but the time of convergence can be extremely long.…”

“…Many fundamental partial differential equations, concerning different areas, exhibit such behavior. Among others, we include viscous shock problems (see, for example [18], [19], [22], [29] for viscous conservation laws, and [6], [33] for Burgers type's equations), relaxation models as the Jin-Xin system [31], phase transition problems described by the Allen-Cahn equation, with the fundamental contributions [8,11] and the most recent references [26,32], and the Cahn-Hilliard equation studied in [1] and [27].…”

“…Starting from there, the strategy of constructing invariant manifolds of approximate steady states have been widely used to describe the slow motion of solutions to different PDEs. We quote here the reference [4], as well as the more recent contributions [22,31,32].…”

Metastability for scalar conservation laws in a bounded domain

“…To name just some of these results, we recall here [5], a pioneering article in the study of slow dynamics for viscous scalar conservation law; starting from this, there are several papers concerning slow motion of internal shock layer for viscous conservation law, see for example [6], [7], [10]. Slow dynamics of interfaces has been examined also for Burgers type equation (see [2] and [12] ), for relaxation system, with the contribution [11], and for phase transition problem described by the Allen-Cahn and Cahn-Hilliard equation in [1,3,4,9].…”

Metastable dynamics for hyperbolic variations of the Allen-Cahn equation

“…A large class of different evolution PDEs, concerning many different areas, exhibits the phenomenon of metastability. Without claiming to be complete, we list some of the principal models that have been analyzed: scalar conservation laws [19,31,32,34,38], the Cahn-Hilliard equation [1,3,4,7,36], Gierer-Meinhardt and Gray-Scott systems [40], Keller-Segel chemotaxis models [16,37], general gradient flows [35], high-order systems [30], gradient systems with equal depth multiple-well potentials [5,6], Cahn-Morral systems [23], the Jin-Xin system [39].…”