959 research outputs found
Singular Cucker-Smale Dynamics
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
Hydrodynamic limit of the kinetic Cucker-Smale flocking model
The hydrodynamic limit of a kinetic Cucker-Smale model is investigated. In
addition to the free-transport of individuals and the Cucker-Smale alignment
operator, the model under consideration includes a strong local alignment term.
This term was recently derived as the singular limit of an alignment operator
due to Motsch and Tadmor. The model is enhanced with the addition of noise and
a confinement potential. The objective of this work is the rigorous
investigation of the singular limit corresponding to strong noise and strong
local alignment. The proof relies on the relative entropy method and entropy
inequalities which yield the appropriate convergence results. The resulting
limiting system is an Euler-type flocking system.Comment: 23 page
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