Confinement in nonlocal interaction equations

J.A. Carrillo; M. Di Francesco; A. Figalli; T. Laurent; D. Slepčev

doi:10.1016/j.na.2011.08.057

Confinement in nonlocal interaction equations

J.A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, D. Slepčev

Department of Mathematical Sciences

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Abstract

We investigate some dynamical properties of nonlocal interaction equations. We consider sets of particles interacting pairwise via a potential W, as well as continuum descriptions of such systems. The typical potentials we consider are repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially localized configuration remains localized for all times, regardless of the number of particles or their arrangement. In particular we find sufficient conditions on the potential W for the above "confinement" property to hold. We use the framework of weak measure solutions developed in Carrillo et al. (2011) [2] to provide unified treatment of both particle and continuum systems.

Original language	English
Pages (from-to)	550-558
Number of pages	9
Journal	Nonlinear Analysis
Volume	75
Issue number	2
DOIs	https://doi.org/10.1016/j.na.2011.08.057
Publication status	Published - 1 Jan 2012

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10.1016/j.na.2011.08.057

Author's accepted version
NOTICE: this is the author’s version of a work that was accepted for publication in Nonlinear Analysis: Theory, Methods & Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Nonlinear Analysis: Theory, Methods & Applications, vol 75, issue 2, 2012, DOI 10.1016/j.na.2011.08.057
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Cite this

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AU - Carrillo, J.A.

AU - Di Francesco, M.

AU - Figalli, A.

AU - Laurent, T.

AU - Slepčev, D.

PY - 2012/1/1

Y1 - 2012/1/1

N2 - We investigate some dynamical properties of nonlocal interaction equations. We consider sets of particles interacting pairwise via a potential W, as well as continuum descriptions of such systems. The typical potentials we consider are repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially localized configuration remains localized for all times, regardless of the number of particles or their arrangement. In particular we find sufficient conditions on the potential W for the above "confinement" property to hold. We use the framework of weak measure solutions developed in Carrillo et al. (2011) [2] to provide unified treatment of both particle and continuum systems.

AB - We investigate some dynamical properties of nonlocal interaction equations. We consider sets of particles interacting pairwise via a potential W, as well as continuum descriptions of such systems. The typical potentials we consider are repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially localized configuration remains localized for all times, regardless of the number of particles or their arrangement. In particular we find sufficient conditions on the potential W for the above "confinement" property to hold. We use the framework of weak measure solutions developed in Carrillo et al. (2011) [2] to provide unified treatment of both particle and continuum systems.

UR - https://www.scopus.com/pages/publications/80055013696

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DO - 10.1016/j.na.2011.08.057

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SN - 0362-546X

VL - 75

SP - 550

EP - 558

JO - Nonlinear Analysis

JF - Nonlinear Analysis

IS - 2

ER -

Confinement in nonlocal interaction equations

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