Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic

Giuseppe Maria Coclite; Nicola De Nitti; Mauro Garavello; Francesca Marcellini; Giuseppe Maria Coclite; Nicola De Nitti; Mauro Garavello; Francesca Marcellini

doi:10.3934/nhm.2021011

Networks and Heterogeneous Media

2021, Volume 16, Issue 3: 413-426. doi: 10.3934/nhm.2021011

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Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic

1.
Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona 4, 70125 Bari, Italy
2.
Department of Data Science, Chair in Applied Analysis (Alexander von Humboldt Professorship), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
3.
Department of Mathematics and its Applications, University of Milano Bicocca, via R. Cozzi 55, 20125 Milano, Italy
4.
Department of Information Engineering, University of Brescia, via Branze 38, 25123 Brescia, Italy

Received: 01 October 2020 Revised: 01 March 2021 Published: 20 May 2021
Primary: 35L65, 35L45, 35B25

We prove the convergence of the vanishing viscosity approximation for a class of $ 2\times2 $ systems of conservation laws, which includes a model of traffic flow in congested regimes. The structure of the system allows us to avoid the typical constraints on the total variation and the $ L^1 $ norm of the initial data. The key tool is the compensated compactness technique, introduced by Murat and Tartar, used here in the framework developed by Panov. The structure of the Riemann invariants is used to obtain the compactness estimates.
- Traffic flow,
- conservation laws,
- hyperbolic systems,
- vanishing viscosity,
- compensated compactness
Citation: Giuseppe Maria Coclite, Nicola De Nitti, Mauro Garavello, Francesca Marcellini. Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic[J]. Networks and Heterogeneous Media, 2021, 16(3): 413-426. doi: 10.3934/nhm.2021011

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Abstract

We prove the convergence of the vanishing viscosity approximation for a class of $ 2\times2 $ systems of conservation laws, which includes a model of traffic flow in congested regimes. The structure of the system allows us to avoid the typical constraints on the total variation and the $ L^1 $ norm of the initial data. The key tool is the compensated compactness technique, introduced by Murat and Tartar, used here in the framework developed by Panov. The structure of the Riemann invariants is used to obtain the compactness estimates.

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