Hydrodynamic limit of the kinetic Cucker–Smale flocking model

Abstract

The hydrodynamic limit of a kinetic Cucker-Smale flocking model is investigated. The starting point is the model considered in [Existence of weak solutions to kinetic flocking models, SIAM Math. Anal. 45 (2013) 215-243], which in addition to free transport of individuals and a standard Cucker-Smale alignment operator, includes Brownian noise and strong local alignment. The latter was derived in [On strong local alignment in the kinetic Cucker-Smale equation, in Hyperbolic Conservation Laws and Related Analysis with Applications (Springer, 2013), pp. 227-242] as the singular limit of an alignment operator first introduced by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 141 (2011) 923-947].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 a relative entropy method. The asymptotic dynamics is described by an Euler-type flocking system.

Asymptotic limit of a spatially-extended mean-field FitzHugh–Nagumo model

“…Our approach to prove the diffusive limit follows ideas from [7,13,17,18], that is we use a relative entropy argument. The specificity of our problem is the absence of noise in the microscopic system (1.4), which implies the absence of a Laplace operator in v in the kinetic equation (1.6).…”

“…The specificity of our problem is the absence of noise in the microscopic system (1.4), which implies the absence of a Laplace operator in v in the kinetic equation (1.6). Hence, without the regularizing effect of noise, the solution f ε of the kinetic equation (1.6) converges towards a Dirac distribution in v, which prevents us from using a more usual entropy of type f logpf q as in [18]. As in [7,13,17], we rather focus on the evolution of moments of second order in v and w of f ε .…”

Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain

“…An interesting feature of the C-S model is that it exhibits a phase-like transition from disordered states to ordered states, depending on the spatial decay rate β of the communication weight ψ. Indeed, Cucker and Smale [10] derived sufficient conditions for global flocking in terms of the initial configuration and communication weight, and these results were further improved in [15,16] (see [18,19,20] for generalized particle and kinetic C-S models). However, when the number of particles is sufficiently large, system (2.1) can be described effectively by the one-particle density function f = f (x, ξ, t) at the spatial velocity position (x, ξ) ∈ R d × R d at time t > 0, which has been obtained in the mean-field limit.…”

Global existence and uniqueness of measure valued solutions to a Vlasov‐type equation with local alignment

“…Also, there are lots of literature about the kinetic equations for CS type models and their hydrodynamic limits. 11,[30][31][32][33][34][35][36] In this paper, we first analyze the unconditional flocking behavior for a weighted MT model (see (6)). We also consider flocking behavior for a model with a "tail" (see (21) and Lemmas 2.1 and 2.2).…”