Microscopic derivation of the one-dimensional constrained Euler equations

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 +, 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.