On the Hydrodynamic Limits of Kinetic Cucker–Smale Model

Abstract

This paper focuses on the hydrodynamic limit problem from a kinetic Cucker–Smale model with a confining potential toward the self-organized hydrodynamic system. This paper aims to establish a uniform-in-$\varepsilon $ convergence rate for the asymptotic expansion ansatz, and hence significantly improves the result obtained in Jiang–Luo–Zhang (Math Models Methods Appl Sci 34:2395–2467, 2024). The key ingredient of the proof is to consider a new decomposition for the remainder function, by handling the contribution of the macroscopic density separately. Moreover, the convergence holds valid in any higher-order Sobolev spaces with the general regularity index.

Keywords

Hydrodynamic limits / Cucker–Smale model / A-priori estimate / Coercivity estimate / 35Q35 / 35Q84 / 35Q92 / 82C22

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Funding

National Natural Science Foundation of China(12371224, 11971360 and 11731008)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature