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Microscopic derivation of the one-dimensional constrained Euler equations

Charlotte Perrin

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.

Microscopic derivation of the one-dimensional constrained Euler equations

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 +, 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.

Paper Structure

This paper contains 17 sections, 23 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Microscopic dynamics
  3. Discretisation of the macroscopic initial datum
  4. Discrete configuration space
  5. Sticky dynamics and reduction to first order
  6. Estimates for the microscopic variables
  7. Second-order formulation and microscopic pressure
  8. Hydrodynamic limit
  9. Interpolation of the microscopic variables
  10. Uniform estimates
  11. The second-order differential inclusion
  12. Eulerian variables and the constrained Euler equations
  13. Karush-Kuhn-Tucker theory
  14. Technical lemmas
  15. Uniqueness issues at the macroscopic level
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(\rho^0,u^0)$ be an initial Eulerian datum such that Let $(X^0,U^0)$ be such that (i) Discretization of the initial datum. Let $N> 1$, and consider $N$ particles of identical radius $r>0$ such that There exists a sequence of positions and velocities $(\mathbf{\rm x}^0_N,\mathbf{\rm u}^0_N) = ((x_i^0)_{i=1,\dots,N}, (u_i^0)_{i=1,\dots,N})$ such that and if $(X_0^N,U_0^N)$ denote the assoc

Theorems & Definitions (47)

  • Theorem 1.1: Main result
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1: Translation of the configuration space
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 37 more