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We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019.

In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by ${ℒ}$, is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker(${ℒ}$) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations.

The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator ${ℒ}$ is not symmetric, i.e. ${ℒ}\ne {{ℒ}}^{\ast }$, where ${{ℒ}}^{\ast }$ is the dual of ${ℒ}$. Moreover, the collision invariants lies in Ker(${{ℒ}}^{\ast }$), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear.

In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

We consider a discrete kinetic approximation of the isentropic Euler equations, and establish the local convergence of the solutions of these relaxation systems to those of the hydrodynamic equations in the hyperbolic limit. We rely on modulated entropy methods and cover the time interval in which the latter admits smooth solutions.