Nonlinear Differential Equations and Applications NoDEA | 2021
Quasi-static limit for a hyperbolic conservation law
Abstract
The term quasi-static evolution refers to dynamics driven by external boundary conditions or forces that change in a time scale much longer than the typical time scale of the convergence to stationary state of the dynamics. In the time scale of the changes of the exterior conditions the system is very close to the corresponding stationary state. This ideal evolutions are fundamental in Thermodynamics and in many other situations. We are interested in studying dynamics where the corresponding quasi-stationary state is of non-equilibrium, i.e. it presents nonvanishing currents of conserved quantities. In a companion article [4] we study the quasi-static limit for the one-dimensional open asymmetric simple exclusion process (ASEP). The symmetric case was studied in [3]. This is a dynamics where the stationary non-equilibrium states are well studied [5, 10, 11]. The macroscopic equation for the ASEP is given by the traffic flow equation on the one-dimensional finite interval r0, 1s: