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.

On an aggregation model with long and short range interactions / M. Burger, V. Capasso, D. Morale. - In: NONLINEAR ANALYSIS: REAL WORLD APPLICATIONS. - ISSN 1468-1218. - 8:3(2007), pp. 939-958. [10.1016/j.nonrwa.2006.04.002]

On an aggregation model with long and short range interactions

V. Capasso
Secondo
;
D. Morale
Ultimo
2007

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.
Aggregation; Nonlinear diffusion; Nonlocal interactions; Swarming behaviour
Settore MAT/06 - Probabilita' e Statistica Matematica
2007
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/24807
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