I am honored to be awarded an NSF CAREER grant (DMS #2238219) on a five-year project: Nonlocal Partial Differential Equations in Collective Dynamics and Fluid Flow. 

Abstract

Collective behaviors are ubiquitous in nature and society. The mathematical study of collective dynamics has been active and fast-growing in recent decades. Many models have been proposed and analyzed to explain the intrinsic nonlocal interactions and the resulting complex emergent phenomena. These models are described by nonlocal partial differential equations. They have deep connections to classical systems in fluid dynamics. The goal of this project is to develop novel and robust analytical techniques to understand the collective behaviors driven by nonlocal structures. The training and professional development of graduate students and young researchers is an integral part of the project.

The project studies three families of partial differential equations with shared nonlocal structures that can affect the solutions of the equations: existence, uniqueness, regularity, and long-time asymptotic behaviors. The first problem is on the compressible Euler system with nonlinear velocity alignment, which describes the remarkable flocking phenomenon in animal swarms. Global phenomena and asymptotic behaviors of the system will be investigated, with a focus on the nonlinearity in the velocity alignment. The second problem is on the pressureless Euler system, aiming at the long-standing question of the uniqueness of weak solutions. The plan is to approximate the system by the relatively well-studied Euler-alignment system in collective dynamics. The third problem is on the Euler-Monge-Ampère system which is closely related to the incompressible Euler equations in fluid dynamics. The embedded nonlocal geometric structure of the system will be explored, with interesting applications in optimal transport and mean-field games.

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 #2238219

McKenzie Black and Changhui Tan

Journal of Differential Equations, Volume 380, pp. 198-227 (2024)

Abstract

We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.

	doi:10.1016/j.jde.2023.10.044
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	This work is supported by NSF grants DMS #2108264 and DMS 2238219
	This work is supported by a UofSC VPR SPARC grant.

Xiang Bai, Qianyun Miao, Changhui Tan and Liutang Xue

Nonlinearity, Volume 37, 025007, 46pp. (2024).

Abstract

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as \(t\to\infty\).

	doi:10.1088/1361-6544/ad140b
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Yi Sun and Changhui Tan

Physica D, Volume 446, 133657 (2023)

Abstract

This paper presents a class of one-dimensional cellular automata (CA) models on traffic flows, featuring nonlocal look-ahead interactions. We develop kinetic Monte Carlo (KMC) algorithms to simulate the dynamics. The standard KMC method can be inefficient for models with global interactions. We design an accelerated KMC method to reduce the computational complexity in the evaluation of the nonlocal transition rates. We investigate several numerical experiments to demonstrate the efficiency of the accelerated algorithm, and obtain the fundamental diagrams of the dynamics under various parameter settings.

	doi:10.1016/j.physd.2023.133657
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

Thomas Hamori and Changhui Tan

Nonlinear Analysis: Real World Applications, Volume 73, 103899, (2023).

Abstract

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.

	doi:10.1016/j.nonrwa.2023.103899
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant