Hydrodynamic Cucker--Smale Model with Normalized Communication Weights and Time Delay

Abstract

We study a hydrodynamic Cucker-Smale-type model with time delay in communication and information processing, in which agents interact with each other through normalized communication weights. The model consists of a pressureless Euler system with time delayed non-local alignment forces. We resort to its Lagrangian formulation and prove the existence of its global in time classical solutions. Moreover, we derive a sufficient condition for the asymptotic flocking behavior of the solutions. Finally, we show the presence of a critical phenomenon for the Eulerian system posed in the spatially onedimensional setting.for t ≥ 0 and x ∈ R d with d ∈ N the space dimension. The constant τ ≥ 0 denotes the fixed delay in communication and information processing. Here and in the sequel we denote by the subscript {·} t the time-dependence of the respective variable. The influence function ψ : R d → R + satisfies the following set of assumptions:

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

“…We refer to the recent reviews [7,14] for the detailed descriptions of the modeling and related literature. There are several works on the pressureless Euler alignment system; the global regularity of classical solutions is obtained in the Eulerian formulation [22] and in the Lagrangian formulation [15,23], and critical thresholds between the supercritical regions with finite-time breakdown and the subcritical region with global-in-time regularity of classical solutions are investigated in one dimension [3,8,33]. More recently, the global regularity for the pressureless fractional Euler alignment system is also established in [19,28,30,31].…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…The hydrodynamic system (1.9) has a rich variety of phenomena compared to the plain pressureless Euler system. This fact is due to the competition between attraction/repulsion and alignment leading to sharp thresholds for the global existence of strong solutions versus finite time blow-up and decay to equilibrium, see [13][14][15]26,63,68]. We emphasize that the additional alignment, linear damping and attraction/repulsion terms can promote the existence of global solutions depending on the intial data.…”

A Simple Proof of Asymptotic Consensus in the Hegselmann--Krause and Cucker--Smale Models with Normalization and Delay

“…In the opposite case the flocking is conditional, i.e., the asymptotic behavior of the system depends on the initial configuration. Cucker-Smale-type models with delay were studied in [17,8,12,2,3]. In particular, in [17] and [2] the precise form (5) with the normalized communication rates (2) was considered (with constant time delay) and asymptotic flocking was proved under a smallness condition on the delay, related to the decay properties of the influence function ψ and the velocity diameter of the initial datum.…”