Publication 10-CNA-20

Confinement in Nonlocal Interaction Equations

J. A. Carrillo
Departament de Matemàtiques
Universitat Autónoma de Barcelona
Bellaterra, Spain
carrillo@mat.uab.es

M. Difrancesco
Dipartimento di Matematica Pura ed Applicata
Università di L'Aquila
L'Aquila, Italy
mdifrance@gmail.com

A. Figalli
Deoartment of Mathematics
University of Texas at Austinn
Austin, TX 78712
figalli@math.utexas.edu

T. Laurent
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
laurent@math.ucla.edu

D. Slepcev
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA
slepcev@andrew.cmu.edu

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 conguration 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 [8] to provide unfied treatment of both particle and continuum systems.

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• 10-CNA-020.pdf

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