Hydrodynamic models of self-organized dynamics: Derivation and existence theory

Abstract

This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attractionrepulsion. The starting point is the kinetic model considered in [10] with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in [10], the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.Acknowledgements: The authors wish to acknowledge the hospitality

Symmetry-breaking phase transitions in highly concentrated semen

“…Apart from the absence of the diffusion terms and of the random noise (those in the last line of (2.10)), the relaxed Euler model (2.7) and (2.9) is similar to the TT model (set P ( n ) = D log n , , , c 2 = 1). Additional diffusion terms such as those of the TT model can also be obtained in the SOH model [45,46]. Here, because we are aiming at a simple qualitative agreement, they are ignored.…”

Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

“…So, we get: 10) which shows that the game with augmented cost function µ f (y) given by (3.3) is a potentiel game associated to potential U. Its variational structure is associated to the following free energy functional (thanks to (3.17) (3.18) and (5.10)):…”

“…We note that because of the presence of the projection P Ω ⊥ the constraint |Ω| = 1 is propagated in time as soon as it is verified at time t = 0. We note that the system is not in conservative form but, in some conditions, it can be shown to be well-posed [10].…”

Spectral method for a kinetic swarming model

“…Local existence and uniqueness have been studied in [12] and we refer to [33] for the implementation of an accurate numerical scheme.…”

“…Several works have already studied kinetic models for swarming [10][11][12], but few have done a numerical investigation. In this work, we first introduce a numerical scheme for a kinetic swarming model based on the Vicsek model.…”