Skip to main content
SpringerLink
Log in

Solvability of Inverse Extremal Problems for Stationary Heat and Mass Transfer Equations

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider inverse extremal problems for the stationary system of heat and mass transfer equations describing the propagation of a substance in a viscous incompressible heat conducting fluid in a bounded domain with Lipschitz boundary. The problems consist in finding some unknown parameters of a medium or source densities from a certain information of a solution. We study solvability of the direct boundary value problem and the inverse extremal problem, justify application of the Lagrange principle, introduce and analyze the optimality systems, and establish sufficient conditions for uniqueness of solutions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andreev V. K., Kaptsov O. V., Pukhnach,ëv V. V., and Rodionov A. A., Application of Group-Theoretic Methods to Hydrodynamics [in Russian], Nauka, Novosibirsk (1994).

    Google Scholar 

  2. Marchuk G. I., Mathematical Modeling in the Problem of Environment [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  3. Belolipetskiî V. M. and Shokin Yu. I., Mathematical Modeling in Environmental Problems [in Russian], Infolio-Press, Novosibirsk (1997).

    Google Scholar 

  4. Gunzburger M. D., Hou L., and Svobodny T. P., “The approximation of boundary control problems for fluid flows with an application to control by heating and cooling,” Comput. Fluids, 22, 239-251 (1993).

    Google Scholar 

  5. Abergel F. and Casas E., “Some optimal control problems of multistate equation appearing in fluid mechanics,” Math. Modeling Numer. Anal., 27, 223-247 (1993).

    Google Scholar 

  6. Alekseev G. V., “Stationary boundary control problems for heat convection equations,” Dokl. Akad. Nauk, 362, No. 2, 174-177 (1998).

    Google Scholar 

  7. Alekseev G. V., “Solvability of stationary boundary control problems for heat convection equations,” Sibirsk. Mat. Zh., 39, No. 5, 982-998 (1998).

    Google Scholar 

  8. Alekseev G. V. and Tereshko D. A., “On solvability of inverse extremal problems for stationary equations of viscous heat conducting fluid,” J. Inverse Ill-posed Problems, 6, No. 6, 521-562 (1998).

    Google Scholar 

  9. Alekseev G. V. and Tereshko D. A., “Stationary optimal control problems for equations of viscous heat conducting fluid,” Sibirsk. Zh. Industr. Mat., 1, No. 2, 24-44 (1998).

    Google Scholar 

  10. Ito K. and Ravindran S. S., “Optimal control of thermally convected fluid flows,” SIAM J. Sci. Comput., 19, No. 6, 1847-1869 (1998).

    Google Scholar 

  11. Alekseev G. V., Inverse Problems of Finding the Admixture Sources in Viscous Fluids [Preprint, No. 8], IPM DVO Ross. Akad. Nauk, Vladivostok (1999).

    Google Scholar 

  12. Cãpãtinã A. and Stavre R., “A control problem in biconvective flow,” J. Math. Kyoto Univ. (JMKYAZ), 37, No. 4, 585-595 (1998).

    Google Scholar 

  13. Adomavichyus, È. A. and Alekseev G. V., Theoretical Analysis of Inverse Extremal Problems for the Stationary Equations of Mass Transfer [Preprint, No. 7], IPM DVO Ross. Akad. Nauk, Vladivostok (1999).

    Google Scholar 

  14. Adomavichyus, È. A. and Alekseev G. V., Theoretical Analysis of Inverse Extremal Problems for the Stationary Equations of Mass Transfer. II [Preprint, No. 18], IPM DVO Ross. Akad. Nauk, Vladivostok (1999).

    Google Scholar 

  15. Ames K. A. and Straughan B., “Stability and Newtons's law of cooling in double diffusive flow,” J. Math. Anal. Appl., 230, 57-69 (1999).

    Google Scholar 

  16. Yang G. Z. and Zabaras N., “The adjoint method for an inverse design problem in the directional solidification of binary alloys,” J. Comp. Phys., 140, No. 2, 432-452 (1998).

    Google Scholar 

  17. Temam R., Navier-Stokes Equations [Russian translation], Mir, Moscow (1981).

    Google Scholar 

  18. Finn R. and Solonnikov V., “Gradient estimates for solutions of the Navier-Stokes equations,” Topol. Methods Nonlinear Anal., 9, 29-39 (1997).

    Google Scholar 

  19. Alekseev G. V., Theoretical Analysis of Inverse Extremal Problems for the Stationary Equations of Heat and Mass Transfer [Preprint, No. 7], IPM DVO Ross. Akad. Nauk, Vladivostok (2000).

    Google Scholar 

  20. Alekseev G. V., “Inverse extremal problems for the stationary equations of heat and mass transfer,” Dokl. Akad. Nauk, 375, No. 3, 315-319 (2000).

    Google Scholar 

  21. Ioffe A. D. and Tikhomirov V. M., The Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  22. Galdi G., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. 2, Springer-Verlag, New York (1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Applied Mathematics of the Far East Division of the Russian Academy of Sciences, Khabarovsk

    G. V. Alekseev

Authors
  1. G. V. Alekseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alekseev, G.V. Solvability of Inverse Extremal Problems for Stationary Heat and Mass Transfer Equations. Siberian Mathematical Journal 42, 811–827 (2001). https://doi.org/10.1023/A:1011940606843

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011940606843

Keywords

Search

Navigation