Time-Dependent Mean-Field Games with Logarithmic Nonlinearities

Handle URI:
http://hdl.handle.net/10754/582932
Title:
Time-Dependent Mean-Field Games with Logarithmic Nonlinearities
Authors:
Gomes, Diogo A. ( 0000-0002-3129-3956 ) ; Pimentel, Edgard
Abstract:
In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Time-Dependent Mean-Field Games with Logarithmic Nonlinearities 2015, 47 (5):3798 SIAM Journal on Mathematical Analysis
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Mathematical Analysis
Issue Date:
6-Oct-2015
DOI:
10.1137/140984622
Type:
Article
ISSN:
0036-1410; 1095-7154
Additional Links:
http://epubs.siam.org/doi/10.1137/140984622
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGomes, Diogo A.en
dc.contributor.authorPimentel, Edgarden
dc.date.accessioned2015-11-30T13:16:14Zen
dc.date.available2015-11-30T13:16:14Zen
dc.date.issued2015-10-06en
dc.identifier.citationTime-Dependent Mean-Field Games with Logarithmic Nonlinearities 2015, 47 (5):3798 SIAM Journal on Mathematical Analysisen
dc.identifier.issn0036-1410en
dc.identifier.issn1095-7154en
dc.identifier.doi10.1137/140984622en
dc.identifier.urihttp://hdl.handle.net/10754/582932en
dc.description.abstractIn this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/140984622en
dc.rightsArchived with thanks to SIAM Journal on Mathematical Analysisen
dc.subjectmean-field gamesen
dc.subjectlogarithmic nonlinearitiesen
dc.subjectnonlinear adjoint methoden
dc.titleTime-Dependent Mean-Field Games with Logarithmic Nonlinearitiesen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalSIAM Journal on Mathematical Analysisen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionDepartment of Mathematics, Universidade Federal de Sao Carlos, 13560 Sao Carlo-SP, Brazilen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorGomes, Diogo A.en
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