Microscopic derivation of the one-dimensional constrained Euler equations

TL;DR

This paper derives one-dimensional constrained Euler equations from microscopic particle models, establishing a rigorous link between particle interactions and macroscopic congestion constraints in fluid flow.

Contribution

It introduces a microscopic particle approximation with inelastic collisions to derive macroscopic congestion constraints in Euler equations.

Findings

Established existence of weak solutions with density constraints.

Connected microscopic Signorini conditions to macroscopic congestion pressure.

Reduced dynamics to a first-order evolution using monotonicity properties.

Abstract

We provide a new existence result for weak solutions to the one-dimensional Euler equations with a maximal density constraint, corresponding to a unilateral constraint on the density. Such models arise in the description of congestion phenomena in compressible flows. Our approach is based on a microscopic approximation by a system of N solid particles of identical radius r, with 2r = 1/N . The particles move freely until collision, after which perfectly inelastic interactions are imposed, so that colliding particles stick together. At this level, the non-overlapping condition is encoded through Signorini-type constraints from contact mechanics. Passing to the limit as N $\rightarrow$ +$\infty$, we rigorously establish the connection between these microscopic Signorini conditions and the macroscopic unilateral constraint on the density, together with the associated sign condition on the congestion pressure. The analysis is carried out in a Lagrangian framework, which is natural at the microscopic level and relies at the macroscopic level on the monotone rearrangement associated with the density. A key ingredient of our result is a monotonicity property of the congested region, which allows us to reduce the dynamics to a first-order evolution in time.