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Stability of Partially Congested Travelling Wave Solutions for the Dissipative Aw-Rascle System.

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|December 22, 2025
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Summary

This study proves the non-linear stability of travelling waves in the dissipative Aw-Rascle system. This extends previous findings to include singular viscosity, crucial for modeling congestion effects in fluid dynamics.

Keywords:
Aw-Rascle systemCompressible Navier-Stokes equationsNonlinear stabilitySingular limitViscous shock waves

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Area of Science:

  • Fluid Dynamics
  • Nonlinear Partial Differential Equations
  • Mathematical Physics

Background:

  • The Aw-Rascle system models fluid dynamics, including congestion.
  • Previous studies on its stability had limitations regarding singular viscosity.
  • Travelling-wave solutions are key to understanding system dynamics.

Purpose of the Study:

  • To prove the non-linear stability of travelling-wave solutions for the dissipative Aw-Rascle system with singular viscosity.
  • To extend existing stability results to cases with singular viscosity.
  • To analyze solutions approximating the 'hard-congestion model'.

Main Methods:

  • Development of carefully weighted energy estimates.
  • Analysis of travelling-wave solutions.
  • Formal equivalence to the compressible pressureless Navier-Stokes system with singular viscosity.

Main Results:

  • Non-linear stability of travelling-wave solutions is proven for the Aw-Rascle system with a singular offset function.
  • The stability holds under small zero integral perturbations.
  • This work extends stability results to systems with singular viscosity.

Conclusions:

  • The non-linear stability of viscous shock waves in the Aw-Rascle system is established, even with singular viscosity.
  • The findings are relevant for approximating solutions to the 'hard-congestion model'.
  • This research advances the understanding of complex fluid flow phenomena.