Research

Papers

McKenzie Black, Changhui Tan. Hydrodynamic Limit Of A Kinetic Flocking Model with Nonlinear Velocity Alignment. (Submitted)

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 [8], 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.

This work has been supported by the NSF grants DMS-2108264 and DMS-2238219.

McKenzie Black, Changhui Tan. Asymptotic behaviors for the compressible Euler system with nonlinear velocity alignment. 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. 

This work is supported by NSF grant DMS #2108264 and the U of SC SPARC grant.