Skip to main content
Springer Nature Link
Log in

Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Abergel, C., Casas, F.: Some optimal control problems of multistate equations appearing in fluid mechanics. Math. Model. Numer. Anal. 27, 223–247 (1993)

    Article  MathSciNet  Google Scholar 

  • Alekseev, G.V.: Solvability of stationary problems of boundary control for thermal convection equations. Sib. Math. J. 39(5), 844–858 (1998)

    Article  MathSciNet  Google Scholar 

  • Ammar-Khodja, F., Santos, M.M.: 2D density-dependent Leray problem with a discontinuous density. Methods Appl. Anal. 13(4), 321–335 (2006)

    MathSciNet  MATH  Google Scholar 

  • Amrouche, C., Penel, P., Seloula, N.: Some remarks on the boundary conditions in the theory of Navier–Stokes equations. Annales Mathématiques Blaise Pascal 20, 37–73 (2013)

    Article  MathSciNet  Google Scholar 

  • Antonsev, S.N., Kazhikov, A.V., Monakov, B.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. North-Holland, Amsterdam (1990)

    Google Scholar 

  • Beirão da Veiga, H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Commun. Pure Appl. Math. 48, 552–577 (2005)

    Article  MathSciNet  Google Scholar 

  • Beirão da Veiga, H.: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip type boundary conditions. Adv. Differ. Equ. 9(9–10), 1079–1114 (2004)

    MathSciNet  MATH  Google Scholar 

  • Borzì, A., Park, E.-J., Vallejos Lass, M.: Multigrid optimization methods for the optimal control of convection diffusion problems with bilinear control. J. Optim. Theory Appl. 168, 510–533 (2016)

    Article  MathSciNet  Google Scholar 

  • Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  • Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2, p. 585. Springer, Berlin (2000)

    Book  Google Scholar 

  • De los Reyes, J.C., Kunisch, K.: A semi-smooth Newton method for control constrained boundary optimal control of the Navier–Stokes equations. Nonlinear Anal. 62, 1289–1316 (2005)

    Article  MathSciNet  Google Scholar 

  • Frolov, N.N.: On the solvability of a boundary value problem of the motion of a nonhomogeneous fluid. Math. Notes 53(5–6), 650–656 (1993)

    Article  MathSciNet  Google Scholar 

  • Frolov, N.N.: A boundary value problem that describes the motion of a nonhomogeneous fluid. Sib. Math. J. 37(2), 376–393 (1996)

    Article  MathSciNet  Google Scholar 

  • Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Res. Inst. Math. Sci. Kōkyūroku 888, 199–216 (1994)

    MathSciNet  MATH  Google Scholar 

  • Fujita, H.: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149, 57–69 (2002)

    Article  MathSciNet  Google Scholar 

  • Gunzburger, M.D., Hou, L., Svobodny, T.P.: Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30(1), 167–181 (1992)

    Article  MathSciNet  Google Scholar 

  • Hettich, R., Kaplan, A., Tichatschke, R.: Regularized penalty methods for ill-posed optimal control problems with elliptic equations. Part I: distributed control with bounded control set and state constraints. Control Cybern. 26, 5–28 (1997)

    MATH  Google Scholar 

  • Hettich, R., Kaplan, A., Tichatschke, R.: Regularized penalty methods for ill-posed optimal control problems with elliptic equations. Part II: distributed control with unbounded set and state constraints. Control Cybern. 26, 29–42 (1997)

    MATH  Google Scholar 

  • Illarionov, A.I.: Optimal boundary control of steady-state flow of a viscous inhomogeneous incompressible fluid. Math. Notes 69(5), 614–624 (2001)

    Article  MathSciNet  Google Scholar 

  • Jägger, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001)

    Article  MathSciNet  Google Scholar 

  • Joseph, D.D.: Fluids dynamics of mixtures of incompressible miscible liquids. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., P’eriaux, J., Pironneau, O. (eds.) Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, vol. 15. Springer, Dordrecht (2010)

    Chapter  Google Scholar 

  • Kazhikov, A.V.: Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid. Dokl. Akad. Nauh. 216, 1008–1010 (1974)

    MathSciNet  Google Scholar 

  • Köner, A., Vexler, B.: A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Mech. 230, 781–802 (2009)

    Article  MathSciNet  Google Scholar 

  • Lee, H.-C., Imanuvilov, O.Y.: Analysis of Neumann boundary optimal control problems for the stationary Boussinesq equations including solid media. SIAM J. Control Optim. 2(39), 457–477 (2000)

    Article  MathSciNet  Google Scholar 

  • Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  Google Scholar 

  • Lions, J.L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)

    Google Scholar 

  • Maxwell, J.C.: On stresses in rarefed gases arising from inequalities of temperature. Philos. Trans. R. Soc. 170, 231–256 (1879)

    Article  Google Scholar 

  • Mallea-Zepeda, E., Ortega-Torres, E., Villamizar-Roa, É.J.: An optimal control problem for the steady nonhomogeneous asymmetric fluids. Appl. Math. Optim. (2017). https://doi.org/10.1007/s00245-017-9466-5

    Article  MathSciNet  Google Scholar 

  • Mulone, G., Salemi, F.: On the existence of hydrodynamic motion in a domain with free boundary type conditions. Meccanica 18(3), 136–144 (1983)

    Article  Google Scholar 

  • Mulone, G., Salemi, F.: On the hydrodynamic motion in a domain with mixed boundary conditions: existence, uniqueness, stability and linearization principle. Ann. Mat. Pura Appl. 139(4), 147–174 (1985)

    Article  MathSciNet  Google Scholar 

  • Navier, C.L.M.H.: Sur le lois de l’équilibrie et du mouvement des corps élastiques. Mem. Acad. R. Sci. Inst. France 369 (1827)

  • Rodríguez-Bellido, M.A., Rueda-Gómez, D.A., Villamizar-Roa, É.J.: On a distributed control problem for a coupled chemotaxis-fluid model. Discret. Contin. Dyn. Syst. Ser. B. 22, 557–571 (2017)

    MathSciNet  MATH  Google Scholar 

  • Santos, M.M.: Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density. Z. Angew. Math. Phys. 53, 661–675 (2002)

    Article  MathSciNet  Google Scholar 

  • Schöberl, J., Simon, R., Zulehner, W.: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49(4), 1482–1503 (2011)

    Article  MathSciNet  Google Scholar 

  • Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure. SIAM J. Math. Anal. 21, 1093–1117 (1990)

    Article  MathSciNet  Google Scholar 

  • Simon, J.: Sur le fluides visqueaux incompressibles et non homogènes. CRAS Paris 309, 447–451 (1989)

    MATH  Google Scholar 

  • Solonnikov, V.A., Scadilov, V.E.: A certain boundary value problem for the stationary system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 125, 196–210 (1973)

    MathSciNet  MATH  Google Scholar 

  • Stavre, R.: A distributed control problem for two coupled fluids in a porous medium. SIAM J. Control Optim. 53(1), 313–335 (2015)

    Article  MathSciNet  Google Scholar 

  • Vallejos Lass, M., Borzì, A.: A robust multigrid method for elliptic optimal control problems. SIAM J. Numer. Anal. 49(4), 1482–1503 (2011)

    Article  MathSciNet  Google Scholar 

  • Verfüth, R.: Finite element approximation of incompressible Navier–Stokes equations with slip boundary condition. Numer. Math. 50, 697–721 (1987)

    Article  MathSciNet  Google Scholar 

  • Vitoriano, F.: On the steady viscous flow of a nonhomogeneous asymmetric fluid. Ann. Mat. Pura Appl. 192, 665–672 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

E. Mallea-Zepeda was supported by Proyecto UTA-Mayor 4740-18, Universidad de Tarapacá, Chile. E. Lenes was supported by the Departamento de Investigaciones of the Universidad del Sinú, Colombia.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad de Tarapacá, Av. 18 de Septiembre 2222, Arica, Chile

    Exequiel Mallea-Zepeda

  2. Departamento de Investigaciones, Universidad del Sinú, Elías Bechara Zainúm, Av. al Bosque Tr. No. 30-729, Cartagena, Colombia

    Eber Lenes

  3. Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile

    Jonnathan Rodríguez Zambrano

  4. Departamento de Matemática, Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile

    Jonnathan Rodríguez Zambrano

Authors
  1. Exequiel Mallea-Zepeda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eber Lenes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonnathan Rodríguez Zambrano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Exequiel Mallea-Zepeda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mallea-Zepeda, E., Lenes, E. & Rodríguez Zambrano, J. Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition. Bull Braz Math Soc, New Series 50, 871–887 (2019). https://doi.org/10.1007/s00574-019-00131-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-019-00131-6

Keywords

Mathematics Subject Classification

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy