On an aggregation model with long and short range interactions |
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| Institution: | 1. Institut für Industriemathematik, Johannes Kepler Universität Linz, Altenbergerstr. 69, A-4040 Linz, Austria;2. ADAMSS (Advanced Applied Mathematical and Statistical Sciences) & Department of Mathematics, Università di Milano, Via Saldini 50, I-20133 Milano, Italy;1. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, PR China;2. Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, PR China;1. Mathematics Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z2;2. Biology Department, Simon Fraser University, Burnaby, BC, Canada V5A 1S6;1. CMUC, Department of Mathematics, University of Coimbra, Portugal;2. Department of Mathematics, University of Coimbra, Portugal;1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy;2. Dipartimento di Ingegneria Civile e Industriale, Largo Lucio Lazzarino, 56122 Pisa, Italy |
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| Abstract: | In recent papers the authors had proposed a stochastic model for swarm aggregation, based on individuals subject to long range attraction and short range repulsion, in addition to a classical Brownian random dispersal. Under suitable laws of large numbers they showed that, for a large number of individuals, the evolution of the empirical distribution of the population can be expressed in terms of an approximating nonlinear degenerate and nonlocal parabolic equation, which describes the limit.In this paper the well-posedness of such evolution equation is investigated, which invokes a notion of entropy solutions extended to the nonlocal case. We motivate entropy solutions from the discrete particle system and use them to prove uniqueness. Moreover, we provide existence results and discuss some basic properties of solutions. Finally, we apply a Lagrangian numerical scheme to perform numerical simulations in spatial dimension one. |
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