Optimal convergence speed in the classical limits of relativistic Cucker-Smale models

TL;DR

This paper provides optimal convergence rate estimates for the classical limit of the relativistic Cucker-Smale model, accounting for relativistic effects and allowing different initial data, improving upon previous constraints.

Contribution

It introduces a novel quantitative estimate with an optimal convergence rate for the classical limit of the relativistic Cucker-Smale model, considering relativistic pressure effects and flexible initial conditions.

Findings

Optimal convergence rate established for classical limit

Relativistic effects included in the pressure term

Initial data constraints relaxed compared to previous work

Abstract

We study quantitative estimates for the flocking and uniform-time classical limit to the relativistic Cucker-Smale (in short RCS) model introduced in \cite{Ha-Kim-Ruggeri-ARMA-2020}. Different from previous works, we do not neglect the relativistic effect on the presence of the pressure in momentum equation. For the RCS model, we provide a quantitative estimate on the uniform-time classical limit with an optimal convergence rate which is the same as in finite-time classical limit under a relaxed initial condition. We also allow corresponding initial data for the RCS and Cucker-Smale (CS) model to be different in the classical limit. This removes earlier constraints employed in the previous classical limit. As a direct application of this optimal convergence rate in the classical limit of the RCS model, we derive an optimal convergence rate for the corresponding uniform-time classical limit for the kinetic RCS model.