Qianyun Miao, Changhui Tan, and Liutang Xue

Mathematical Models and Methods in Applied Sciences, Volume 31, No 3, pp. 473-524 (2021).

Abstract

We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, fea- turing strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment.

	doi:10.1142/S021820252150010X
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	This work is supported by NSF grant DMS #1853001

I have been awarded a grant from the Office of the Vice President for Research at the University of South Carolina, on a one-year project: Multiscale nonlocal models in traffic flows.

Project Summary

Mathematical models on traffic flows have been studied extensively in the past century. Many celebrated models lie in a beautiful multiscale framework. The investigations on these models play an important role in designing traffic networks and preventing traffic jams.

Recently, the fast development of self-driving vehicles and new communication technologies allow long-range interactions in traffic networks. It attracts a lot of interest in nonlocal traffic flow models.

The aim of the proposed project is to develop the mathematical theory on non-local traffic flows and understand how nonlocal interactions can help to optimize the traffic networks and avoid the creation of traffic congestions.

The PI has been actively working on a variety of multiscale nonlocal models in physical, biological, and sociological contexts. These experiences can greatly help the understanding of the nonlocal phenomena in traffic models. Preliminary investigations show intriguing behaviors and promising outcomes. Results generated from this project will be capitalized to prepare proposals for external grants from NSF and DOT.

UofSC Office of Research Awards Announcement Page

Yi Sun, and Changhui Tan

Physica D, Volume 413, 132663 (2020).

Abstract

This paper presents a new class of one-dimensional (1D) traffic models with look-ahead rules that take into account of two effects: nonlocal slow-down effect and right-skewed non-concave asymmetry in the fundamental diagram. The proposed 1D cellular automata (CA) models with the Arrhenius type look-ahead interactions implement stochastic rules for cars’ movement following the configuration of the traffic ahead of each car. In particular, we take two different look-ahead rules: one is based on the distance from the car under consideration to the car in front of it; the other one depends on the car density ahead. Both rules feature a novel idea of multiple moves, which plays a key role in recovering the non-concave flux in the macroscopic dynamics. Through a semi-discrete mesoscopic stochastic process, we derive the coarse-grained macroscopic dynamics of the CA model. We also design a numerical scheme to simulate the proposed CA models with an efficient list-based kinetic Monte Carlo (KMC) algorithm. Our results show that the fluxes of the KMC simulations agree with the coarse-grained macroscopic averaged fluxes for the different look-ahead rules under various parameter settings.

	doi:10.1016/j.physd.2020.132663
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	This work is supported by NSF grant DMS #1853001
	This work is supported by a UofSC VPR ASPIRE I grant

Yongki Lee, and Changhui Tan

Communications in Mathematical Sciences, Volume 20, No. 4, pp. 1151-1172 (2022).

Abstract

We study a Lighthill-Whitham-Richards (LWR) type traffic flow model, with a nonlocal look-ahead interaction that has a slow-down effect depending on the traffic ahead. We show a sharp critical threshold condition on the initial data that distinguishes global smooth solutions and finite- time wave breakdown. It is well-known that the LWR model leads to a finite-time shock formation, representing the creation of traffic jams, for generic smooth initial data with finite mass. Our result shows that the nonlocal slowdown effect can help to prevent shock formations, for a class of subcritical initial data.

	doi:10.4310/CMS.2022.v20.n4.a9
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	This work is supported by NSF grants DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

Changhui Tan

Nonlinearity, Volume 33, No 4, pp. 1907-1924 (2020).

Abstract

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of the Cucker–Smale model on animal flocks. For the Euler-alignment system with bounded interactions, a critical threshold phenomenon is proved in Tadmor and Tan (2014 Phil. Trans. R. Soc. A 372 20130401), where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in Do et al (2018 Arch. Ration. Mech. Anal. 228 1–37), for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

	doi:10.1088/1361-6544/ab6c39
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	This work is supported by NSF grant DMS #1853001