LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY
Autor(a) principal: | |
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Data de Publicação: | 2011 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1090/S0002-9947-2010-05206-7 http://hdl.handle.net/11449/22168 |
Resumo: | In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case. |
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LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITYIn this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Univ Estadual Campinas UNICAMP, Dept Matemat, IMECC, BR-13083859 Campinas, SP, BrazilUniv Estadual Paulista UNESP, Dept Matemat, BR-15054000 Sj do Rio Preto, SP, BrazilUniv Estadual Paulista UNESP, Dept Matemat, BR-15054000 Sj do Rio Preto, SP, BrazilFAPESP: 07/51490-7FAPESP: 01/06984-5CNPq: 303301/2007-4CNPq: 302214/2004-6Amer Mathematical SocUniversidade Estadual de Campinas (UNICAMP)Universidade Estadual Paulista (Unesp)Lopes Filho, Milton C.Nussenzveig Lopes, Helena J.Precioso, Juliana C. [UNESP]2014-05-20T14:02:55Z2014-05-20T14:02:55Z2011-05-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article2641-2661application/pdfhttp://dx.doi.org/10.1090/S0002-9947-2010-05206-7Transactions of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 363, n. 5, p. 2641-2661, 2011.0002-9947http://hdl.handle.net/11449/2216810.1090/S0002-9947-2010-05206-7WOS:000290511300014WOS000290511300014.pdfWeb of Sciencereponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengTransactions of the American Mathematical Society1.4962,378info:eu-repo/semantics/openAccess2023-11-09T06:11:40Zoai:repositorio.unesp.br:11449/22168Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462023-11-09T06:11:40Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
title |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
spellingShingle |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY Lopes Filho, Milton C. |
title_short |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
title_full |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
title_fullStr |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
title_full_unstemmed |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
title_sort |
LEAST ACTION PRINCIPLE and THE INCOMPRESSIBLE EULER EQUATIONS WITH VARIABLE DENSITY |
author |
Lopes Filho, Milton C. |
author_facet |
Lopes Filho, Milton C. Nussenzveig Lopes, Helena J. Precioso, Juliana C. [UNESP] |
author_role |
author |
author2 |
Nussenzveig Lopes, Helena J. Precioso, Juliana C. [UNESP] |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Universidade Estadual de Campinas (UNICAMP) Universidade Estadual Paulista (Unesp) |
dc.contributor.author.fl_str_mv |
Lopes Filho, Milton C. Nussenzveig Lopes, Helena J. Precioso, Juliana C. [UNESP] |
description |
In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case. |
publishDate |
2011 |
dc.date.none.fl_str_mv |
2011-05-01 2014-05-20T14:02:55Z 2014-05-20T14:02:55Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1090/S0002-9947-2010-05206-7 Transactions of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 363, n. 5, p. 2641-2661, 2011. 0002-9947 http://hdl.handle.net/11449/22168 10.1090/S0002-9947-2010-05206-7 WOS:000290511300014 WOS000290511300014.pdf |
url |
http://dx.doi.org/10.1090/S0002-9947-2010-05206-7 http://hdl.handle.net/11449/22168 |
identifier_str_mv |
Transactions of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 363, n. 5, p. 2641-2661, 2011. 0002-9947 10.1090/S0002-9947-2010-05206-7 WOS:000290511300014 WOS000290511300014.pdf |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Transactions of the American Mathematical Society 1.496 2,378 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
2641-2661 application/pdf |
dc.publisher.none.fl_str_mv |
Amer Mathematical Soc |
publisher.none.fl_str_mv |
Amer Mathematical Soc |
dc.source.none.fl_str_mv |
Web of Science reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
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1803046397739532288 |