Soft congestion approximation to the one-dimensional constrained Euler equations

TL;DR

This paper analyzes the one-dimensional compressible Euler equations with a singular pressure law, establishing existence of solutions and rigorously justifying the limit towards free-congested Euler equations, highlighting the impact of congestion on solution behavior.

Contribution

It provides the first rigorous analysis of the singular limit from compressible to free-congested Euler equations with a detailed description of congestion effects.

Findings

Existence of bounded weak solutions via viscous regularization.

Detailed description of solution breakdown due to singular pressure.

Rigorous justification of the limit towards free-congested Euler equations.

Abstract

This article is concerned with the analysis of the one-dimensional compressible Euler equations with a singular pressure law, the so-called hard sphere equation of state. The result is twofold. First, we establish the existence of bounded weak solutions by means of a viscous regularization and refined compensated compactness arguments. Second, we investigate the smooth setting by providing a detailed description of the impact of the singular pressure on the breakdown of the solutions. In this smooth framework, we rigorously justify the singular limit towards the free-congested Euler equations, where the compressible (free) dynamics is coupled with the incompressible one in the constrained (i.e. congested) domain.