Isothermal hydrodynamic limit for kinetic flocking models with nonlinear velocity alignment
Abstract
We study the hydrodynamic limit of kinetic flocking models with nonlinear velocity alignment and strong local Fokker--Planck forcing. In the limit of small Knudsen number the kinetic densities converge to a local Maxwellian centered around macroscopic quantities satisfying the compressible Euler system with isothermal pressure p = s ρ, while macroscopic alignment is obtained by convolution of the microscopic velocity law with the standard Gaussian, resulting in a different nonlinearity. Such discrepancy has been observed already in \cite{black2025hydrodynamic}. In this note we rigorously justify the limit for admissible weak kinetic solutions and non-vacuous limiting macroscopic solution. Our analysis provides a quantitative rate of convergence in terms of relative entropy: cH(fe | μ) ≲ e provided initially cH(fe(0) | μ(0)) ≤ e. In the linear case we recover the known result \cite{karper2015hydrodynamic}.
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BibTeX
@misc{shvydkoy2026isothermal,
title = {{Isothermal hydrodynamic limit for kinetic flocking models with nonlinear velocity alignment}},
author = {Roman Shvydkoy and Changhui Tan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30457},
eprint = {2609.30457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30457},
}
RIS
TY - UNPB
TI - Isothermal hydrodynamic limit for kinetic flocking models with nonlinear velocity alignment
AU - Roman Shvydkoy
AU - Changhui Tan
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.30457
UR - https://arxiv.org/abs/2609.30457
AB - We study the hydrodynamic limit of kinetic flocking models with nonlinear velocity alignment and strong local Fokker--Planck forcing. In the limit of small Knudsen number the kinetic densities converge to a local Maxwellian centered around macroscopic quantities satisfying the compressible Euler system with isothermal pressure p = s ρ, while macroscopic alignment is obtained by convolution of the microscopic velocity law with the standard Gaussian, resulting in a different nonlinearity. Such discrepancy has been observed already in \cite{black2025hydrodynamic}. In this note we rigorously justify the limit for admissible weak kinetic solutions and non-vacuous limiting macroscopic solution. Our analysis provides a quantitative rate of convergence in terms of relative entropy: cH(f_e | μ) ≲ e provided initially cH(f_e(0) | μ(0)) ≤ e. In the linear case we recover the known result \cite{karper2015hydrodynamic}.
ER -