Xiang Bai, Changhui Tan and Liutang Xue

Journal of Differential Equations, Volume 407, pp. 269-310 (2024).

Abstract

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter \(\alpha\in(0,1)\). This highlights the distinct feature of the alignment operator.

	doi:10.1016/j.jde.2024.06.020
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

McKenzie Black, and Changhui Tan

Kinetic and Related Models, Volume 18, No. 4, pp. 609-632 (2025)

Abstract

We investigate a class of Vlasov-type kinetic flocking models featuring nonlinear velocity alignment. Our primary objective is to rigorously derive the hydrodynamic limit leading to the compressible Euler system with nonlinear alignment. This study builds upon the work by Figalli and Kang [Anal. PDE, 12(3), 843-866, 2018], which addressed the scenario of linear velocity alignment using the relative entropy method. The introduction of nonlinearity gives rise to an additional discrepancy in the alignment term during the limiting process. To effectively handle this discrepancy, we employ the monokinetic ansatz in conjunction with the relative entropy approach. Furthermore, our analysis reveals distinct nonlinear alignment behaviors between the kinetic and hydrodynamic systems, particularly evident in the isothermal regime.

	doi:10.3934/krm.2024028
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

Yatao Li, Qianyun Miao, Changhui Tan and Liutang Xue

International Mathematics Research Notices, Volume 2024, No. 23, pp. 14393-14422 (2024).

Abstract

We investigate global solutions to the Euler-alignment system in d dimensions with unidirectional flows and strongly singular communication protocols \(\phi(x)=|x|^{-d+\alpha}\) for \(\alpha\in(0,2)\). Our paper establishes global regularity results in both the subcritical regime \(1<\alpha<2\) and the critical regime \(\alpha=1\). Notably, when \(\alpha=1\), the system exhibits a critical scaling similar to the critical quasi-geostrophic equation. To achieve global well-posedness, we employ a novel method based on propagating the modulus of continuity. Our approach introduces the concept of simultaneously propagating multiple moduli of continuity, which allows us to effectively handle the system of two equations with critical scaling. Additionally, we improve the regularity criteria for solutions to this system in the supercritical regime \(0<\alpha<1\).

	doi:10.1093/imrn/rnae246
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

Trevor M. Leslie and Changhui Tan

Journal of Evolution Equations, Volume 24, Article 8, 45pp. (2024).

Abstract

We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated to a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated to a scalar balance law formulation of the system.

	doi:10.1007/s00028-023-00939-2
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	This work is supported by NSF grants DMS #2108264 and DMS #2238219

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