STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS

Handle URI:
http://hdl.handle.net/10754/599721
Title:
STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS
Authors:
FELLNER, KLEMENS; RAOUL, GAËL
Abstract:
In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states ρ̄ ε approximate weakly a unique stationary state ρ̄ ∈ L∞. We illustrate our results with numerical examples. © 2010 World Scientific Publishing Company.
Citation:
FELLNER K, RAOUL G (2010) STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS. Mathematical Models and Methods in Applied Sciences 20: 2267–2291. Available: http://dx.doi.org/10.1142/S0218202510004921.
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Dec-2010
DOI:
10.1142/S0218202510004921
Type:
Article
ISSN:
0218-2025; 1793-6314
Sponsors:
Both authors would like to thank Prof. Christian Schmeiser for initiating the research done in this paper and Dr. Marco Di Francesco for many valuable discussions. K.F. has been supported by Award No. KUK-I1-007-43 of Peter A. Markowich, made by King Abdullah University of Science and Technology (KAUST) and by the bilateral Austria-France project (Austria: FR 05/2007 France: Amadeus 13785 UA). G.R. has been partially supported by the DEASE program affiliated at the WPI, Wolfgang Pauli Institute, University of Vienna.
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorFELLNER, KLEMENSen
dc.contributor.authorRAOUL, GAËLen
dc.date.accessioned2016-02-28T06:08:18Zen
dc.date.available2016-02-28T06:08:18Zen
dc.date.issued2010-12en
dc.identifier.citationFELLNER K, RAOUL G (2010) STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS. Mathematical Models and Methods in Applied Sciences 20: 2267–2291. Available: http://dx.doi.org/10.1142/S0218202510004921.en
dc.identifier.issn0218-2025en
dc.identifier.issn1793-6314en
dc.identifier.doi10.1142/S0218202510004921en
dc.identifier.urihttp://hdl.handle.net/10754/599721en
dc.description.abstractIn this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states ρ̄ ε approximate weakly a unique stationary state ρ̄ ∈ L∞. We illustrate our results with numerical examples. © 2010 World Scientific Publishing Company.en
dc.description.sponsorshipBoth authors would like to thank Prof. Christian Schmeiser for initiating the research done in this paper and Dr. Marco Di Francesco for many valuable discussions. K.F. has been supported by Award No. KUK-I1-007-43 of Peter A. Markowich, made by King Abdullah University of Science and Technology (KAUST) and by the bilateral Austria-France project (Austria: FR 05/2007 France: Amadeus 13785 UA). G.R. has been partially supported by the DEASE program affiliated at the WPI, Wolfgang Pauli Institute, University of Vienna.en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectdouble-well potentialen
dc.subjectNon-local interaction equationen
dc.subjectnumerical simulationen
dc.subjectstability analysisen
dc.titleSTABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONSen
dc.typeArticleen
dc.identifier.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionEcole Normale Superieure de Cachan, Cachan, Franceen
kaust.grant.numberKUK-I1-007-43en
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