Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations

Handle URI:
http://hdl.handle.net/10754/597985
Title:
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Authors:
Lorz, Alexander; Mirrahimi, Sepideh; Perthame, Benoît
Abstract:
Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.
Citation:
Lorz A, Mirrahimi S, Perthame B (2011) Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations. Communications in Partial Differential Equations 36: 1071–1098. Available: http://dx.doi.org/10.1080/03605302.2010.538784.
Publisher:
Informa UK Limited
Journal:
Communications in Partial Differential Equations
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
17-Jan-2011
DOI:
10.1080/03605302.2010.538784
Type:
Article
ISSN:
0360-5302; 1532-4133
Sponsors:
This research is supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorLorz, Alexanderen
dc.contributor.authorMirrahimi, Sepidehen
dc.contributor.authorPerthame, Benoîten
dc.date.accessioned2016-02-25T13:10:23Zen
dc.date.available2016-02-25T13:10:23Zen
dc.date.issued2011-01-17en
dc.identifier.citationLorz A, Mirrahimi S, Perthame B (2011) Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations. Communications in Partial Differential Equations 36: 1071–1098. Available: http://dx.doi.org/10.1080/03605302.2010.538784.en
dc.identifier.issn0360-5302en
dc.identifier.issn1532-4133en
dc.identifier.doi10.1080/03605302.2010.538784en
dc.identifier.urihttp://hdl.handle.net/10754/597985en
dc.description.abstractNonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses coexist? We will explain how these questions relate to the so-called "constrained Hamilton-Jacobi equation" and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution. © Taylor & Francis Group, LLC.en
dc.description.sponsorshipThis research is supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherInforma UK Limiteden
dc.subjectAdaptive evolutionen
dc.subjectDirac concentrationsen
dc.subjectHamilton-Jacobi equationen
dc.subjectLotka-Volterra parabolic equationen
dc.titleDirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equationsen
dc.typeArticleen
dc.identifier.journalCommunications in Partial Differential Equationsen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionUniversite Pierre et Marie Curie, Paris, Franceen
kaust.grant.numberKUK-I1-007-43en
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