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

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 $C_b^{1,α}$, 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.

Figures (3)

• Figure 1: From the left top, mass, velocity momentum, energy, and entropy. Although the velocity moment looks like it is decreasing, the difference is not that big compare to it's value.
• Figure 2: Profile of the function $h(t,v)$ with $\gamma=0.1,\ 1.0,\text{ and } 5.0$. As $t$ increases the profile's amplitude goes higher. This agrees with the alignment property of our equation (in momentum or energy), since it means that the distribution aligns in velocity.
• Figure 3: Each final distribution is produced by $\gamma=0.1,\ 5.0$, respectively.

Theorems & Definitions (31)

• Remark 1.1
• Proposition 2.1
• proof
• Remark 2.2: Entropy sign and comparison
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Theorem 3.1
• proof