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Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system

Nilasis Chaudhuri, Muhammed Ali Mehmood, Charlotte Perrin, Ewelina Zatorska

Abstract

We introduce the notion of duality solution for the hard-congestion model on the real line, and additionally prove an existence result for this class of solutions. Our study revolves around the analysis of a generalised Aw-Rascle system, where the offset function is replaced by the gradient of a singular function, such as $ρ$ $γ$ n , where $γ$ $\rightarrow$ $\infty$. We prove that under suitable assumptions on the initial data, solutions to the Aw-Rascle system converge towards the so-called duality solutions, which have previously found applications in other systems which exhibit compressive dynamics. We also prove that one can obtain weak solutions to the limiting system under stricter assumptions on the initial data. Finally, we discuss (non-)uniqueness issues.

Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system

Abstract

We introduce the notion of duality solution for the hard-congestion model on the real line, and additionally prove an existence result for this class of solutions. Our study revolves around the analysis of a generalised Aw-Rascle system, where the offset function is replaced by the gradient of a singular function, such as n , where . We prove that under suitable assumptions on the initial data, solutions to the Aw-Rascle system converge towards the so-called duality solutions, which have previously found applications in other systems which exhibit compressive dynamics. We also prove that one can obtain weak solutions to the limiting system under stricter assumptions on the initial data. Finally, we discuss (non-)uniqueness issues.
Paper Structure (19 sections, 26 theorems, 112 equations)

This paper contains 19 sections, 26 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Duality solutions
  3. Main results
  4. Outline of the paper
  5. Global existence of regular solutions for Lg fixed
  6. Local existence of regular solutions
  7. Energy estimates
  8. The blow-up lemma
  9. Proving the existence of a global solution
  10. Uniform estimates
  11. Estimates for the momentum
  12. Estimates for the velocity Lg
  13. Estimates for the potential Lg
  14. The limit passage
  15. Existence of duality solutions to the hard-congestion model
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $T>0$ and let $(\rho^0,m^0 ,\pi^0)$ be such that Then there exists a duality solution $(\rho, m = \rho u + \partial_x \pi, \pi)$ to the hard-congestion model in the sense of Definition defndualityHCL below.

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 1.2: Reversible solutions
  • Definition 1.3: Generalised backward flow bouchut1998one
  • Definition 1.4: Duality solutions
  • Remark 1.5
  • Theorem 1.6: Bouchut and James bouchut1998one
  • Definition 1.7
  • Definition 1.8: Regular solutions
  • Remark 1.9
  • Theorem 1.10: Global existence for $n$ fixed
  • ...and 48 more