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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Two-dimensional pseudo-gravity model: Particles motion in a non-potential singular force field

Authors: Julien Barré, Dan Crisan and Thierry Goudon
Journal: Trans. Amer. Math. Soc. 371 (2019), 2923-2962
MSC (2010): Primary 82C70, 70F45, 35Q35
Published electronically: November 27, 2018
MathSciNet review: 3896101
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Abstract: We analyze a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self-consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation. We establish the existence of solutions under a suitable smallness condition on the total mass or, equivalently, for a sufficiently large diffusion coefficient. When a symmetry assumption is fulfilled, the solutions satisfy strengthened estimates (exponential moments). We also investigate the convergence of the $ N$-particles description towards the PDE system in the mean field regime.

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Additional Information

Julien Barré
Affiliation: Institut Denis Poisson, Université d’Orléans, Université de Tours et CNRS, rue de Chartres, 45067 Orléans Cedex 2, France—and—Institut Universitaire de France

Dan Crisan
Affiliation: Department of Mathematics, Imperial College London, Huxley Building, 180 Queens Gate, London SW7 2BZ, United Kingdom

Thierry Goudon
Affiliation: Université Côte d’Azur, Inria, CNRS, LJAD, Parc Valrose, 06100 Nice, France

Keywords: Attractive forces, convection-diffusion equations, mean field regime
Received by editor(s): February 20, 2017
Received by editor(s) in revised form: February 26, 2018
Published electronically: November 27, 2018
Article copyright: © Copyright 2018 American Mathematical Society