Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model

Young Pil Choi; Seok Bae Yun

doi:10.1137/20M1355914

Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model

Young Pil Choi, Seok Bae Yun

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We study a Paveri-Fontana type model, which describes the evolution of the mesoscopic distribution of vehicles through a combined effect of adjustment of the velocity with respect to nearby vehicles, and slowing down and speeding up of the vehicles arising as a result of exchange of velocity with the vehicles on the same location on the road. We first prove the global-in-time existence of weak solutions. The proof is via energy, L^p, and compact support estimates together with a velocity averaging lemma. The combined effect of the alignment nature of Qr, which keeps the characteristic from spreading, and the dissipative nature of Q_i, which gives the uniform control on the size of the distribution function, is crucially used in the estimates. We also rigorously establish a hydrodynamic limit to the pressureless Euler equation by employing the relative entropy combined with the Monge-Kantorovich-Rubinstein distance.

Original language	English
Pages (from-to)	2631-2659
Number of pages	29
Journal	SIAM Journal on Mathematical Analysis
Volume	53
Issue number	2
DOIs	https://doi.org/10.1137/20M1355914
Publication status	Published - 2021

Bibliographical note

Funding Information:
\ast Received by the editors July 28, 2020; accepted for publication (in revised form) February 2, 2021; published electronically April 29, 2021. https://doi.org/10.1137/20M1355914 Funding: The work of the first author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP), 2017R1C1B2012918, a POSCO Science Fellowship of the POSCO TJ Park Foundation, and Yonsei University Research Fund 2019-22-02. The work of the second author was supported by Samsung Science and Technology Foundation project SSTF-BA1801-02. \dagger Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea (ypchoi@yonsei.ac.kr). \ddagger Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea (sbyun01@skku.edu).

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

All Science Journal Classification (ASJC) codes

Analysis
Computational Mathematics
Applied Mathematics

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10.1137/20M1355914

Cite this

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title = "Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model",

abstract = "We study a Paveri-Fontana type model, which describes the evolution of the mesoscopic distribution of vehicles through a combined effect of adjustment of the velocity with respect to nearby vehicles, and slowing down and speeding up of the vehicles arising as a result of exchange of velocity with the vehicles on the same location on the road. We first prove the global-in-time existence of weak solutions. The proof is via energy, Lp, and compact support estimates together with a velocity averaging lemma. The combined effect of the alignment nature of Qr, which keeps the characteristic from spreading, and the dissipative nature of Qi, which gives the uniform control on the size of the distribution function, is crucially used in the estimates. We also rigorously establish a hydrodynamic limit to the pressureless Euler equation by employing the relative entropy combined with the Monge-Kantorovich-Rubinstein distance.",

author = "Choi, {Young Pil} and Yun, {Seok Bae}",

note = "Funding Information: \ast Received by the editors July 28, 2020; accepted for publication (in revised form) February 2, 2021; published electronically April 29, 2021. https://doi.org/10.1137/20M1355914 Funding: The work of the first author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP), 2017R1C1B2012918, a POSCO Science Fellowship of the POSCO TJ Park Foundation, and Yonsei University Research Fund 2019-22-02. The work of the second author was supported by Samsung Science and Technology Foundation project SSTF-BA1801-02. \dagger Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea (ypchoi@yonsei.ac.kr). \ddagger Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea (sbyun01@skku.edu). Publisher Copyright: Copyright {\textcopyright} by SIAM. Unauthorized reproduction of this article is prohibited.",

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PY - 2021

Y1 - 2021

N2 - We study a Paveri-Fontana type model, which describes the evolution of the mesoscopic distribution of vehicles through a combined effect of adjustment of the velocity with respect to nearby vehicles, and slowing down and speeding up of the vehicles arising as a result of exchange of velocity with the vehicles on the same location on the road. We first prove the global-in-time existence of weak solutions. The proof is via energy, Lp, and compact support estimates together with a velocity averaging lemma. The combined effect of the alignment nature of Qr, which keeps the characteristic from spreading, and the dissipative nature of Qi, which gives the uniform control on the size of the distribution function, is crucially used in the estimates. We also rigorously establish a hydrodynamic limit to the pressureless Euler equation by employing the relative entropy combined with the Monge-Kantorovich-Rubinstein distance.

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Existence and hydrodynamic limit for a Paveri-Fontana type kinetic traffic model

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