Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

Abstract. We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

Abstract. We consider well-posedness of the aggregation equation, in dimensions two and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x| α , α > 2 − d, and prove local well-posedness infor sufficiently large p > p s . In the special case of K(x) = |x|, the exponent p s = d/(d − 1) is sharp for local well-posedness, in that solutions can instantaneously concentrate mass for initial data inWe also give an Osgood condition on the potential K(x) which guaranties global existence and uniqueness in

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A Fokker-Planck type equation for interacting particles with exclusion principle is analyzed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L 1 are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropydissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate.

RESUMEN: En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo ABSTRACT:In this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology research.

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We consider an agent-based model of emotional contagion coupled with motion in one dimension that has recently been studied in the computer science community. The model involves movement with a speed proportional to a "fear" variable that undergoes a temporal consensus averaging based on distance to other agents. We study the effect of Riemann initial data for this problem, leading to shock dynamics that are studied both within the agent-based model as well as in a continuum limit. We examine the behavior of the model under distinguished limits as the characteristic contagion interaction distance and the interaction timescale both approach zero. The limiting behavior is related to a classical model for pressureless gas dynamics with "sticky" particles. In comparison, we observe a threshold for the interaction distance vs. interaction timescale that produce qualitatively different behavior for the system -in one case particle paths do not cross and there is a natural Eulerian limit involving nonlocal interactions and in the other case particle paths can cross and one may consider only a kinetic model in the continuum limit.