Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917650754437120 |
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| author | Mehmood, Muhammed Ali |
| author_facet | Mehmood, Muhammed Ali |
| contents | 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 $ρ_{n}^γ$, where $γ\to \infty$. 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, $W_{n} = ρ_{n}^{-1}\partial_{x}w_{n}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01379 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function Mehmood, Muhammed Ali Analysis of PDEs 35Q35, 35B25, 76T20, 90B20 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 $ρ_{n}^γ$, where $γ\to \infty$. 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, $W_{n} = ρ_{n}^{-1}\partial_{x}w_{n}$. 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. |
| title | Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function |
| topic | Analysis of PDEs 35Q35, 35B25, 76T20, 90B20 |
| url | https://arxiv.org/abs/2306.01379 |