Singular Cucker–Smale Dynamics

Abstract

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared -analytically and numerically -to the porous medium equation.

On the Mathematics of Swarming: Emergent Behavior in Alignment Dynamics

“…We conclude by mentioning existence results for the important class of singular kernels ( ) ≔ − with < + 2: in this case, the communication framework emphasizes short-range interactions over long-range interactions, yet their global support still reflects global communication. For the regularity for 1D weakly singular kernels, with < 1, and strongly singular kernels with 1 ≤ < 3 we refer to [MMPZ19] and the references therein. Here, alignment is structured as fractional diffusion which was shown, at least in the one-dimensional case, to enforce unconditional flocking behavior, independent of any initial threshold.…”

“…in which communication heavily emphasizes near-by neighbors over those farther away, e.g., [MMPZ19] and the references therein, and communication based on various random-based protocols found in chemo-and phototactic dynamics, the Elo rating system, voter and related opinion-based models, a random-batch method, and consensus-based optimization, to name but a few. 1 In addition to alignment, pairwise interactions may include repulsions and attractions,…”

On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

“…In closing, we remark that the value s = 1 2 plays a distinguished role in the theory for weakly singular kernels at the discrete and the kinetic levels (in any dimension), at least in terms of what can be proven about the equations. See the works [15,[17][18][19]. Future research may reveal that this role is a technical rather than a fundamental one; nevertheless, it would be interesting to understand whether there is a connection between the reasons this barrier appears at the particle/kinetic and at the hydrodynamic levels.…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…Throughout this paper, we assume that W and ψ satisfy W (x) = W (−x) and ψ(x) = ψ(−x) for x ∈ R d . They include basic particle models for collective behaviors, see [12,20,25,34,36,46,47,63] and the references therein. Our main goal is to derive the macroscopic collective models rigorously governing the evolution of the particle system (1.1) as the number of particles goes to infinity.…”