On the kinetic equation arising from the large-scale limit of the Cucker-Smale model

TL;DR

This paper derives a new kinetic equation from the Cucker-Smale model using a large-scale scaling approach, revealing local spatial alignment and analyzing solutions, well-posedness, and blow-up scenarios.

Contribution

It introduces a novel kinetic equation with local spatial alignment from the Cucker-Smale model and analyzes its solutions and properties.

Findings

Explicit solution for spatially homogeneous case

Global well-posedness for small interaction strength

Finite-time blow-up scenarios for mono-kinetic solutions

Abstract

We propose a large-scale scaling viewpoint for deriving mesoscopic dynamics from interacting particle systems and apply it to the Cucker--Smale flocking model. In contrast with the classical mean-field regime leading to the Vlasov-type Cucker--Smale equation with spatially nonlocal (convolution) alignment force, our scaling yields a kinetic equation whose alignment field becomes local in space and nonlocal only in velocity. For the spatially homogeneous case, we obtain an explicit solution and derive quantitative flocking rates. For the spatially inhomogeneous equation we establish a local well-posedness in $W^{1,\infty}$ and in $C_b^{1,\alpha}$, highlighting the additional difficulties caused by the absence of a convolution structure. Moreover, for sufficiently small interaction strength we present a global well-posedness and a forward-in-time $L^1$ asymptotic completeness property. Finally, we investigate mono-kinetic solutions and exhibit finite-time blow-up scenarios.