Analysis and Derivation of the Compressible Euler System with Nonlinear Velocity Alignment: An Investigation Into Collective Behavior

Collective behavior refers to the observable patterns, actions, or movements that emerge within a group of entities, such as individuals, animals, or particles, when they interact with each other. In this study, we concentrate on the pressureless compressible Euler system, expanding it to incorporate a family of nonlinear velocity alignment. This extension represents a nonlinear iteration of the Euler-alignment system within collective dynamics, revealing intriguing asymptotic emergent phenomena such as alignment and flocking. Our investigation explores different types of nonlinearity and nonlocal communication protocols, unveiling a diverse spectrum of asymptotic behaviors within the system. In focusing on the derivation, we delve into a specific class of Vlasov-type kinetic flocking models characterized by nonlinear velocity alignment. Our primary objective is to systematically derive the hydrodynamic limit, leading to the compressible Euler system with nonlinear alignment. Building upon the foundational work by Figalli and Kang \cite{figalli2018rigorous}, which focused on linear velocity alignment through the relative entropy method, our investigation introduces nonlinearity, resulting in an additional alignment term discrepancy during the limiting process. To address this challenge effectively, we employ the monokinetic ansatz in conjunction with the relative entropy approach. Furthermore, our analysis reveals distinctive nonlinear alignment behaviors, particularly evident in the isothermal regime, highlighting significant differences between the kinetic and hydrodynamic systems.