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

I have been awarded an NSF grant (DMS #1815667) on a three-year project: Regularity and Singularity Formation in Swarming and Related Fluid Models. It is translated to DMS #1853001 when I move to University of South Carolina.

Abstract

Swarming is a commonly observed complex biological and sociological phenomenon. The internal interaction mechanism attracts a lot of attention in physics, engineering, biology, and social sciences. This project is devoted to developing a unified mathematical theory towards the understanding of the swarming dynamics, as well as other nonlocal models that share similar structures. These models are widely considered in fluid mechanics, meteorology, astrophysics, biology, and ecology. The study of the regularity and singularity formations of these equations will provide a firm theoretical foundation for these applications, and also help consolidate the validity of these models in describing the natural phenomena.

The research will focus on understanding the nonlinear and nonlocal phenomena in swarming dynamics, and models having related structures in fluid mechanic. Three different but related models will be investigated. The first model is the Euler-Alignment system, which describes the flocking behavior in animal swarms. The goal is to develop a robust toolbox to analyze the nonlocal alignment operator and its balance with the drift nonlinearity. Similar behaviors are also observed in other fluid equations including porous medium flow, and surface quasi-geostrophic equations, which will be investigated using the same analytical techniques. The second model is the 2D inviscid Boussinesq equations. The global regularity is one of the outstanding problems in fluid dynamics. The idea is to construct solutions to capture the possible singularity formation, starting from some modified versions of the equations. The third model is the kinetic swarming system. The aim is to investigate the important relation between the kinetic equation and a variety of hydrodynamic limits. In particular, different alignment operators will be considered at the kinetic level. They are expected to lead to different macroscopic limits. All these three sub-projects will advance the mathematical understanding of nonlocal PDEs and related applications. They will also provide education and training to graduate and undergraduate students in this active field.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

NSF award page on the grant DMS #1853001

Changhui Tan

Communications in Mathematical Sciences, Volume 17, No 7, pp. 1779-1794 (2019).

Abstract

We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of quasi-geostrophic equation, and also a special case of the Euler alignment system. For strictly positive smooth initial data, global regularity has been proved in [Do, Kiselev, Ryzhik and Tan, Arch. Ration. Mech. Anal., 228(1):1–37, 2018]. We construct a family of non-negative smooth initial data so that solution is not \(C^1\)-uniformly bounded. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.

	doi:10.4310/CMS.2019.v17.n7.a2
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	This work is supported by NSF grant DMS #1853001