等温流体动力学极限：具有非线性速度对齐的动力学群集模型
Isothermal hydrodynamic limit for kinetic flocking models with nonlinear velocity alignment
- University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
- University of South Carolina（南卡罗来纳大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文严格证明了具有非线性速度对齐的动力学群集模型在小Knudsen数极限下收敛到等温可压缩Euler系统，并给出了基于相对熵的定量收敛速率。
AI中文摘要：
我们研究了具有非线性速度对齐和强局部Fokker-Planck强迫的动力学群集模型的流体动力学极限。在小Knudsen数极限下，动力学密度收敛到以宏观量为中心的局部Maxwellian分布，这些宏观量满足具有等温压力 $p = \s \rho$ 的可压缩Euler系统，而宏观对齐是通过将微观速度定律与标准高斯分布卷积得到的，导致不同的非线性性。这种差异已在\cite{black2025hydrodynamic}中观察到。在本文中，我们严格证明了对于可容许的弱动力学解和非真空极限宏观解的极限。我们的分析提供了相对熵意义上的定量收敛速率：$\cH(f_\e | \mu) \lesssim \e\bigl(1+|\log\e|^{p-2}\bigr)$，前提是初始时 $\cH(f_\e(0) | \mu(0)) \leq \e$，其中 $p$ 是对齐力中非线性的阶数。在线性情形 $p=2$ 时，我们恢复了已知结果 \cite{karper2015hydrodynamic}。
英文摘要：
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: $\cH(f_\e | μ) \lesssim \e\bigl(1+|\log\e|^{p-2}\bigr)$ provided initially $\cH(f_\e(0) | μ(0)) \leq \e$, where $p$ is the order of non-linearity in the alignment force. In the linear case $p=2$ we recover the known result \cite{karper2015hydrodynamic}.
