References & Citations
Mathematics > Analysis of PDEs
Title: Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function
(Submitted on 2 Jun 2023 (v1), last revised 26 Apr 2024 (this version, v2))
Abstract: We study the Aw-Rascle system in a one-dimensional domain with periodic boundary conditions, where the offset function is replaced by the gradient of the function $\rho_{n}^{\gamma}$, where $\gamma \to \infty$. The resulting system resembles the 1D pressureless compressible Navier-Stokes system with a vanishing viscosity coefficient in the momentum equation and can be used to model traffic and suspension flows. We first prove the existence of a unique global-in-time classical solution for $n$ fixed. Unlike the previous result for this system, we obtain global existence without needing to add any approximation terms to the system. This is by virtue of a $n-$uniform lower bound on the density which is attained by carrying out a maximum-principle argument on a suitable potential, $W_{n} = \rho_{n}^{-1}\partial_{x}w_{n}$. Then, we prove the convergence to a weak solution of a hybrid free-congested system as $n \to \infty$, which is known as the hard-congestion model.
Submission history
From: Muhammed Mehmood [view email][v1] Fri, 2 Jun 2023 09:00:59 GMT (43kb)
[v2] Fri, 26 Apr 2024 13:38:38 GMT (35kb)
Link back to: arXiv, form interface, contact.