TY - JOUR

T1 - Cucker-Smale model with finite speed of information propagation: Well-posedness, flocking and mean-field limit

AU - Haskovec, Jan

N1 - KAUST Repository Item: Exported on 2022-10-13
Acknowledgements: The author acknowledges the fruitful discussions with Oliver Tse that have taken place during his visit of TU Eindhoven, and with Jan Vyb´ıral during his visit of Czech Technical University in Prague, which helped to initiate and develop some ideas presented in this pape

PY - 2022/9

Y1 - 2022/9

N2 - We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed [Math Processing Error]. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than [Math Processing Error], then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed [Math Processing Error] such that if [Math Processing Error], the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of [Math Processing Error] is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.

AB - We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed [Math Processing Error]. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than [Math Processing Error], then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed [Math Processing Error] such that if [Math Processing Error], the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of [Math Processing Error] is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.

UR - http://hdl.handle.net/10754/678005

UR - https://www.aimsciences.org/article/doi/10.3934/krm.2022033

U2 - 10.3934/krm.2022033

DO - 10.3934/krm.2022033

M3 - Article

JO - Kinetic and Related Models

JF - Kinetic and Related Models

SN - 1937-5093

ER -