First-order aggregation models with alignment

Abstract

We include alignment interactions in a well-studied first-order attractive-repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a secondorder kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…As mentioned before, similar hydrodynamic limits of vanishing inertia type have been considered in recent literature for other related systems like the Vlasov-Poisson-Fokker-Planck, the aggregation equation, the alignment-aggregation system and some other anisotropic versions of the aggregation equation, see 95,96,102,103,110,185 and last Subsection 5.6. Before ending this part, we will sketch the idea for a different suitable system where one can apply such method.…”

“…Such arguments can be made rigorous for Lipschitz-continuous forces via Tikhonov's theorem 145 . See 95,96,102,103,110,185,188 for some recent advances in this line both for smooth and singular kernels at the microscopic and macroscopic levels.…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…At the formal level, the equations (1.11) will be replaced by (1.4)-(1.5) as ε → 0. The limiting nonlinearly coupled aggregation equations (1.4)-(1.5) have been recently studied in [39,40]. Several authors have studied particular choices of interactions V, W and comunication functions ψ for some of the connecting asymptotic limits from the kinetic description (1.10) with/without noise to the hydrodynamic system (1.11) in [8,11,42,57], from the hydrodynamic system (1.11) to the aggregation equation (1.4)- (1.5) in [23,59,60], and for the direct limit from the kinetic equation to the aggregation equation (1.4)- (1.5) in [8,53].…”

“…Remark 1.4. One may follow a similar argument as in [40,Theorem 2.4] to have the existence and uniqueness of classical solutions (ρ,ū) to the equations (1.4)-(1.5) satisfying the regularity properties and assumptions of Theorem 1.2. For the Coulomb or Riesz interaction, an idea of proof proposed in [28] would be employed to establish the local-in-time existence and uniqueness of classical solutions to the equations (1.4)-(1.5) without the confinement potential.…”

Small inertia regularization of an anisotropic aggregation model

“…In [12] the authors study the continuum analogue of (1.5) (with g ≡ 1, i.e., the isotropic version) and show that its solutions converge as ε → 0 to solutions of the PDE counterpart of the first-order model (1.1). Furthermore, [13] considers a generalization, where an additional alignment term is included in the equation for velocity. In this sense, the present paper complements these works, by investigating the zero inertia limit at the discrete/ODE level, and also with the caveat of allowing the interactions to be anisotropic.…”