On The Mean Field limit for Cucker-Smale models

Roberto Natalini (1) , Thierry Paul (2)

1 Consiglio Nazionale delle Ricerche c/o Department of Mathematics
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [12]. Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [8] in the Hamiltonian setting, we show the limit providing explicit constants. %Using an Eulerian point of view introduced in \cite{gmp} in the Hamiltonian setting, we don't use empirical measures and provide explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.

Domaines: Equations aux dérivées partielles [math.AP]; Biologie animale; Sociologie; Physique mathématique [math-ph]; Sciences du Vivant [q-bio]

Roberto Natalini, Thierry Paul. On The Mean Field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - Series B, In press, ⟨10.3934/dcdsb.2021164⟩. ⟨hal-02008699v5⟩