Yatao Li, Qianyun Miao, Changhui Tan and Liutang Xue

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 \(0<\alpha<1\).

This work is supported by NSF grants DMS #2108264 and DMS #2238219

Trevor M. Leslie and Changhui Tan

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.

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

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

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 \(t\to\infty\).

This work is supported by NSF grant DMS #1853001 and DMS #2108264