Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the fl...

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Autores principales:	Evgenii S. Baranovskii, Eber Lenes, Exequiel Mallea-Zepeda, Jonnathan Rodríguez, Lautaro Vásquez
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	Stokes equations variable density variable viscosity mixed boundary conditions Navier slip condition control problems Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/8b6e40dc678a44b6bad2815853e37fba
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id	oai:doaj.org-article:8b6e40dc678a44b6bad2815853e37fba
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spelling	oai:doaj.org-article:8b6e40dc678a44b6bad2815853e37fba2021-11-25T19:06:20ZControl Problem Related to 2D Stokes Equations with Variable Density and Viscosity10.3390/sym131120502073-8994https://doaj.org/article/8b6e40dc678a44b6bad2815853e37fba2021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2050https://doaj.org/toc/2073-8994We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.Evgenii S. BaranovskiiEber LenesExequiel Mallea-ZepedaJonnathan RodríguezLautaro VásquezMDPI AGarticleStokes equationsvariable densityvariable viscositymixed boundary conditionsNavier slip conditioncontrol problemsMathematicsQA1-939ENSymmetry, Vol 13, Iss 2050, p 2050 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Stokes equations variable density variable viscosity mixed boundary conditions Navier slip condition control problems Mathematics QA1-939
spellingShingle	Stokes equations variable density variable viscosity mixed boundary conditions Navier slip condition control problems Mathematics QA1-939 Evgenii S. Baranovskii Eber Lenes Exequiel Mallea-Zepeda Jonnathan Rodríguez Lautaro Vásquez Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
description	We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.
format	article
author	Evgenii S. Baranovskii Eber Lenes Exequiel Mallea-Zepeda Jonnathan Rodríguez Lautaro Vásquez
author_facet	Evgenii S. Baranovskii Eber Lenes Exequiel Mallea-Zepeda Jonnathan Rodríguez Lautaro Vásquez
author_sort	Evgenii S. Baranovskii
title	Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
title_short	Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
title_full	Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
title_fullStr	Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
title_full_unstemmed	Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity
title_sort	control problem related to 2d stokes equations with variable density and viscosity
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/8b6e40dc678a44b6bad2815853e37fba
work_keys_str_mv	AT evgeniisbaranovskii controlproblemrelatedto2dstokesequationswithvariabledensityandviscosity AT eberlenes controlproblemrelatedto2dstokesequationswithvariabledensityandviscosity AT exequielmalleazepeda controlproblemrelatedto2dstokesequationswithvariabledensityandviscosity AT jonnathanrodriguez controlproblemrelatedto2dstokesequationswithvariabledensityandviscosity AT lautarovasquez controlproblemrelatedto2dstokesequationswithvariabledensityandviscosity
_version_	1718410288868360192

Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

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