Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - M. Di Francesco - A. Figalli - T. Laurent - D. SlepĨev

Confinement in nonlocal interaction equations

created by figalli on 28 Aug 2011


Accepted Paper

Inserted: 28 aug 2011

Journal: Nonlinear Anal.
Year: 2011


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 a previous paper to provide unified treatment of both particle and continuum systems.