Hard-congestion limit of the p-system in the BV setting

TL;DR

This paper rigorously justifies the hard congestion limit in a compressible system with singular pressure, demonstrating convergence towards a mixed model in BV solutions for small data, focusing on a single interface front.

Contribution

It provides the first convergence result for the hard congestion limit in BV solutions, using a weighted Glimm functional for front-tracking approximation.

Findings

Convergence of the compressible system to the mixed model for a single interface.

Use of a weighted Glimm functional for BV estimates.

Framework applicable to small data perturbations.

Abstract

This note is concerned with the rigorous justification of the so-called hard congestion limit from a compressible system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics, for small data in the framework of BV solutions. We present a first convergence result for perturbations of a reference state represented by a single propagating large interface front, while the study of a more general framework where the reference state is constituted by multiple interface fronts is announced in the conclusion and will be the subject of a forthcoming paper. A key element of the proof is the use of a suitably weighted Glimm functional that allows to obtain precise estimates on the BV norm of the front-tracking approximation.