Sensitivity equations for measure-valued solutions to transport equations

Autores
Ackleh, Azmy S.; Saintier, Nicolas Bernard Claude; Skrzeczkowski, Jakub
Año de publicación
2020
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).
Fil: Ackleh, Azmy S.. State University of Louisiana; Estados Unidos
Fil: Saintier, Nicolas Bernard Claude. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Skrzeczkowski, Jakub. University of Warsaw; Polonia
Materia
DIFFERENTIABILITY OF SOLUTIONS
SPACE OF RADON MEASURES
TRANSPORT EQUATIONS
VERY WEAK SOLUTIONS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/150450

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network_name_str CONICET Digital (CONICET)
spelling Sensitivity equations for measure-valued solutions to transport equationsAckleh, Azmy S.Saintier, Nicolas Bernard ClaudeSkrzeczkowski, JakubDIFFERENTIABILITY OF SOLUTIONSSPACE OF RADON MEASURESTRANSPORT EQUATIONSVERY WEAK SOLUTIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).Fil: Ackleh, Azmy S.. State University of Louisiana; Estados UnidosFil: Saintier, Nicolas Bernard Claude. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Skrzeczkowski, Jakub. University of Warsaw; PoloniaAmerican Institute of Mathematical Sciences2020-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/150450Ackleh, Azmy S.; Saintier, Nicolas Bernard Claude; Skrzeczkowski, Jakub; Sensitivity equations for measure-valued solutions to transport equations; American Institute of Mathematical Sciences; Mathematical Biosciences And Engineering; 17; 1; 1-2020; 514-5371547-1063CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.aimspress.com/fileOther/PDF/MBE/mbe-17-01-028.pdfinfo:eu-repo/semantics/altIdentifier/doi/10.3934/mbe.2020028info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:18:29Zoai:ri.conicet.gov.ar:11336/150450instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:18:29.541CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Sensitivity equations for measure-valued solutions to transport equations
title Sensitivity equations for measure-valued solutions to transport equations
spellingShingle Sensitivity equations for measure-valued solutions to transport equations
Ackleh, Azmy S.
DIFFERENTIABILITY OF SOLUTIONS
SPACE OF RADON MEASURES
TRANSPORT EQUATIONS
VERY WEAK SOLUTIONS
title_short Sensitivity equations for measure-valued solutions to transport equations
title_full Sensitivity equations for measure-valued solutions to transport equations
title_fullStr Sensitivity equations for measure-valued solutions to transport equations
title_full_unstemmed Sensitivity equations for measure-valued solutions to transport equations
title_sort Sensitivity equations for measure-valued solutions to transport equations
dc.creator.none.fl_str_mv Ackleh, Azmy S.
Saintier, Nicolas Bernard Claude
Skrzeczkowski, Jakub
author Ackleh, Azmy S.
author_facet Ackleh, Azmy S.
Saintier, Nicolas Bernard Claude
Skrzeczkowski, Jakub
author_role author
author2 Saintier, Nicolas Bernard Claude
Skrzeczkowski, Jakub
author2_role author
author
dc.subject.none.fl_str_mv DIFFERENTIABILITY OF SOLUTIONS
SPACE OF RADON MEASURES
TRANSPORT EQUATIONS
VERY WEAK SOLUTIONS
topic DIFFERENTIABILITY OF SOLUTIONS
SPACE OF RADON MEASURES
TRANSPORT EQUATIONS
VERY WEAK SOLUTIONS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).
Fil: Ackleh, Azmy S.. State University of Louisiana; Estados Unidos
Fil: Saintier, Nicolas Bernard Claude. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Skrzeczkowski, Jakub. University of Warsaw; Polonia
description We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).
publishDate 2020
dc.date.none.fl_str_mv 2020-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/150450
Ackleh, Azmy S.; Saintier, Nicolas Bernard Claude; Skrzeczkowski, Jakub; Sensitivity equations for measure-valued solutions to transport equations; American Institute of Mathematical Sciences; Mathematical Biosciences And Engineering; 17; 1; 1-2020; 514-537
1547-1063
CONICET Digital
CONICET
url http://hdl.handle.net/11336/150450
identifier_str_mv Ackleh, Azmy S.; Saintier, Nicolas Bernard Claude; Skrzeczkowski, Jakub; Sensitivity equations for measure-valued solutions to transport equations; American Institute of Mathematical Sciences; Mathematical Biosciences And Engineering; 17; 1; 1-2020; 514-537
1547-1063
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.aimspress.com/fileOther/PDF/MBE/mbe-17-01-028.pdf
info:eu-repo/semantics/altIdentifier/doi/10.3934/mbe.2020028
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv American Institute of Mathematical Sciences
publisher.none.fl_str_mv American Institute of Mathematical Sciences
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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