We investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse (resp. attract) each other in the short (resp. long) range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. © 2012 Elsevier B.V. All rights reserved.
Bibliographical noteKAUST Repository Item: Exported on 2021-03-31
Acknowledged KAUST grant number(s): KUK-I1- 007-43
Acknowledgements: DB and JAC were supported by the projects Ministerio de Ciencia e InnovaciónMTM2011-27739-C04-02 and 2009-SGR-345 from Agència de Gestió d’Ajuts Universitaris i de Recerca-Generalitat de Catalunya. JAC acknowledges the support from the Royal Society through a Wolfson Research Merit Award. GR was supported by Award No. KUK-I1- 007-43 of Peter A. Markowich, made by King Abdullah University of Science and Technology (KAUST). DB, JAC and GR acknowledge partial support from CBDif-Fr ANR-08-BLAN-0333-01 project. TL acknowledges the support from NSF Grant DMS-1109805. This work was supported by Engineering and Physical Sciences Research Council grant number EP/K008404/1. The authors warmly thank Joaquín Pérez in helping them with the differential geometry question related to Lemma 8.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Condensed Matter Physics