Quasi-static limit for a hyperbolic conservation law

TL;DR

This paper investigates the behavior of solutions to one-dimensional hyperbolic conservation laws with time-dependent boundary conditions, focusing on the quasi-static limit where solutions evolve slowly and are governed by stationary profiles.

Contribution

It provides a rigorous analysis of the quasi-static limit for scalar hyperbolic equations with convex or concave flux and time-dependent boundary data, linking dynamic solutions to stationary profiles.

Findings

Characterization of the quasi-static limit for entropy solutions

Demonstration that solutions converge to stationary profiles over time

Analysis applicable to equations with convex or concave flux functions

Abstract

We study the quasi-static limit for the $L^\infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with the quasi-static equation, whose entropy solution is determined by the stationary profile corresponding to the boundary data at a given time.