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      A well-posedness theory in measures for some kinetic models of collective motion

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          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.

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          Novel type of phase transition in a system of self-driven particles

          A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation (\(\eta\)) added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, \(| {\bf v}_a | =0\)) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since \(| {\bf v}_a |\) is found to scale as \((\eta_c-\eta)^\beta\) with \(\beta\simeq 0.45\).
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            Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse

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              Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups

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                Author and article information

                Journal
                22 July 2009
                2010-04-12
                Article
                10.1142/S0218202511005131
                0907.3901
                4eab72ad-d525-4bc6-8c13-de0fa10e0d41

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                35A05, 35B40, 82D99, 92D50
                Mathematical Models and Methods in Applied Sciences, Vol. 21, No. 3, pp. 515-539 (March 2011)
                math.AP

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