On the global existence for the Muskat problem

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law (???) which is satisfied...

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Detalles Bibliográficos
Autores: Constantin, Peter, Córdoba Gazolaz, Diego, Gancedo García, Francisco, Strain, Robert M.
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2013
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/45145
Acceso en línea:http://hdl.handle.net/11441/45145
https://doi.org/10.4171/JEMS/360
Access Level:acceso abierto
Palabra clave:Porous media
Incompressible flows
Fluid interface
Global existence
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spelling On the global existence for the Muskat problemConstantin, PeterCórdoba Gazolaz, DiegoGancedo García, FranciscoStrain, Robert M.Porous mediaIncompressible flowsFluid interfaceGlobal existenceThe Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law (???) which is satisfied by the equation (???) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy ∥f0∥L∞<∞ and ∥∂xf0∥L∞<1. We take advantage of the fact that the bound ∥∂xf0∥L∞<1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ∥f∥1≤1/5. Previous results of this sort used a small constant ϵ≪1 which was not explicit.National Science FoundationMinisterio de Ciencia e InnovaciónEuropean Research CouncilEuropean Mathematical SocietyAnálisis MatemáticoFQM104: Analisis Matematico2013info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/45145https://doi.org/10.4171/JEMS/360reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of the European Mathematical Society, 15 (1), 201-227.DMS-0804380MTM2008-03754StG-203138CDSIFDMS-0901810DMS-0901463http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=15&iss=1&rank=7info:eu-repo/semantics/openAccessoai:idus.us.es:11441/451452026-06-17T12:51:07Z
dc.title.none.fl_str_mv On the global existence for the Muskat problem
title On the global existence for the Muskat problem
spellingShingle On the global existence for the Muskat problem
Constantin, Peter
Porous media
Incompressible flows
Fluid interface
Global existence
title_short On the global existence for the Muskat problem
title_full On the global existence for the Muskat problem
title_fullStr On the global existence for the Muskat problem
title_full_unstemmed On the global existence for the Muskat problem
title_sort On the global existence for the Muskat problem
dc.creator.none.fl_str_mv Constantin, Peter
Córdoba Gazolaz, Diego
Gancedo García, Francisco
Strain, Robert M.
author Constantin, Peter
author_facet Constantin, Peter
Córdoba Gazolaz, Diego
Gancedo García, Francisco
Strain, Robert M.
author_role author
author2 Córdoba Gazolaz, Diego
Gancedo García, Francisco
Strain, Robert M.
author2_role author
author
author
dc.contributor.none.fl_str_mv Análisis Matemático
FQM104: Analisis Matematico
dc.subject.none.fl_str_mv Porous media
Incompressible flows
Fluid interface
Global existence
topic Porous media
Incompressible flows
Fluid interface
Global existence
description The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law (???) which is satisfied by the equation (???) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy ∥f0∥L∞<∞ and ∥∂xf0∥L∞<1. We take advantage of the fact that the bound ∥∂xf0∥L∞<1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ∥f∥1≤1/5. Previous results of this sort used a small constant ϵ≪1 which was not explicit.
publishDate 2013
dc.date.none.fl_str_mv 2013
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/45145
https://doi.org/10.4171/JEMS/360
url http://hdl.handle.net/11441/45145
https://doi.org/10.4171/JEMS/360
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of the European Mathematical Society, 15 (1), 201-227.
DMS-0804380
MTM2008-03754
StG-203138CDSIF
DMS-0901810
DMS-0901463
http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=15&iss=1&rank=7
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv European Mathematical Society
publisher.none.fl_str_mv European Mathematical Society
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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