Weak time regularity and uniqueness for a Q-Tensor model

The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals. The existence of weak solutions and a uniqueness criteria have been already studied (see [11] Marius Paicu and Arghir Zarnescu. Energy dissipation and regularity for a cou...

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Authors: Guillén González, Francisco Manuel, Rodríguez Bellido, María Ángeles
Format: article
Status:Published version
Publication Date:2014
Country:España
Institution:Universidad de Sevilla (US)
Repository:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/40265
Online Access:http://hdl.handle.net/11441/40265
https://doi.org/10.1137/13095015X
Access Level:Open access
Keyword:Q-Tensor
Navier-Stokes equations
regularity
uniqueness
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spelling Weak time regularity and uniqueness for a Q-Tensor modelGuillén González, Francisco ManuelRodríguez Bellido, María ÁngelesQ-TensorNavier-Stokes equationsregularityuniquenessThe coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals. The existence of weak solutions and a uniqueness criteria have been already studied (see [11] Marius Paicu and Arghir Zarnescu. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal., 203(1):45–67, 2012) for a Cauchy problem in the whole R3 and [7] F. Guillén-Gonz´alez and M. A. Rodríguez-Bellido. Weak solutions for an initialboundary Q-tensor problem related to liquid crystals. Submitted, 2014 for a initial-boundary problem in a bounded domain Ω). Nevertheless, results on strong regularity have only been treated in [11] for a Cauchy problem in the whole R3. In this paper, imposing Dirichlet or Neumann boundary conditions, we show the existence and uniqueness of a local in time weak solution with weak regularity for the time derivative of the velocity and the tensor variables (u, Q). Moreover, we gives a regularity criteria implying that this solution is global in time. Note that the regularity furnished by the weak regularity for (u, Q) and the weak regularity for (∂tu, ∂tQ) is not equivalent to the strong regularity. Finally, when large enough viscosity is imposed, we obtain the existence (and uniqueness) of global in time strong solution. In fact, if non-homogeneous Dirichlet condition for Q is imposed, the strong regularity needs to be obtained together with the weak regularity for (∂tu, ∂tQ).Ministerio de Ciencia e Innovación (España) MTM2009-12927Ministerio de Economía y Competitividad (España) MTM2012-32325Society for Industrial and Applied MathematicsEcuaciones Diferenciales y Análisis NuméricoMinisterio de Ciencia e Innovación (MICIN). EspañaMinisterio de Economía y Competitividad (MINECO). España2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/40265https://doi.org/10.1137/13095015Xreponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésSIAM journal on mathematical analysis, 46(5), 3540-3567MTM2009-12927info:eu-repo/grantAgreement/MINECO/MTM2012-32325/info:eu-repo/semantics/openAccessoai:idus.us.es:11441/402652026-06-17T12:51:07Z
dc.title.none.fl_str_mv Weak time regularity and uniqueness for a Q-Tensor model
title Weak time regularity and uniqueness for a Q-Tensor model
spellingShingle Weak time regularity and uniqueness for a Q-Tensor model
Guillén González, Francisco Manuel
Q-Tensor
Navier-Stokes equations
regularity
uniqueness
title_short Weak time regularity and uniqueness for a Q-Tensor model
title_full Weak time regularity and uniqueness for a Q-Tensor model
title_fullStr Weak time regularity and uniqueness for a Q-Tensor model
title_full_unstemmed Weak time regularity and uniqueness for a Q-Tensor model
title_sort Weak time regularity and uniqueness for a Q-Tensor model
dc.creator.none.fl_str_mv Guillén González, Francisco Manuel
Rodríguez Bellido, María Ángeles
author Guillén González, Francisco Manuel
author_facet Guillén González, Francisco Manuel
Rodríguez Bellido, María Ángeles
author_role author
author2 Rodríguez Bellido, María Ángeles
author2_role author
dc.contributor.none.fl_str_mv Ecuaciones Diferenciales y Análisis Numérico
Ministerio de Ciencia e Innovación (MICIN). España
Ministerio de Economía y Competitividad (MINECO). España
dc.subject.none.fl_str_mv Q-Tensor
Navier-Stokes equations
regularity
uniqueness
topic Q-Tensor
Navier-Stokes equations
regularity
uniqueness
description The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals. The existence of weak solutions and a uniqueness criteria have been already studied (see [11] Marius Paicu and Arghir Zarnescu. Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system. Arch. Ration. Mech. Anal., 203(1):45–67, 2012) for a Cauchy problem in the whole R3 and [7] F. Guillén-Gonz´alez and M. A. Rodríguez-Bellido. Weak solutions for an initialboundary Q-tensor problem related to liquid crystals. Submitted, 2014 for a initial-boundary problem in a bounded domain Ω). Nevertheless, results on strong regularity have only been treated in [11] for a Cauchy problem in the whole R3. In this paper, imposing Dirichlet or Neumann boundary conditions, we show the existence and uniqueness of a local in time weak solution with weak regularity for the time derivative of the velocity and the tensor variables (u, Q). Moreover, we gives a regularity criteria implying that this solution is global in time. Note that the regularity furnished by the weak regularity for (u, Q) and the weak regularity for (∂tu, ∂tQ) is not equivalent to the strong regularity. Finally, when large enough viscosity is imposed, we obtain the existence (and uniqueness) of global in time strong solution. In fact, if non-homogeneous Dirichlet condition for Q is imposed, the strong regularity needs to be obtained together with the weak regularity for (∂tu, ∂tQ).
publishDate 2014
dc.date.none.fl_str_mv 2014
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/40265
https://doi.org/10.1137/13095015X
url http://hdl.handle.net/11441/40265
https://doi.org/10.1137/13095015X
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv SIAM journal on mathematical analysis, 46(5), 3540-3567
MTM2009-12927
info:eu-repo/grantAgreement/MINECO/MTM2012-32325/
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 Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
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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