Journal of Mathematical Fluid Mechanics

, Volume 13, Issue 1, pp 33–53

On an Iterative Method for Approximate Solutions of a Generalized Boussinesq Model

  • José Luiz Boldrini
  • Blanca Climent-Ezquerra
  • María Drina Rojas-Medar
  • Marko A. Rojas-Medar
Article

Abstract

An iterative method is proposed for finding approximate solutions of an initial and boundary value problem for a nonstationary generalized Boussinesq model for thermally driven convection of fluids with temperature dependent viscosity and thermal conductivity. Under certain conditions, it is proved that such approximate solutions converge to a solution of the original problem; moreover, convergence-rate bounds for the constructed approximate solutions are also obtained.

Mathematics Subject Classification (2000)

Primary 35Q30 Secondary 76D03 76M99 65M15 

Keywords

Boussinesq equations strong solutions iterative method 

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References

  1. 1.
    Amrouche C., Girault V.: On the existence and regularity of the solutions of Stokes Problem an arbitrary dimension. Proc. Japan Acad. Sect. A 67, 171–175 (1991)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981)Google Scholar
  3. 3.
    Diaz J.I., Galiano G.: On the Boussinesq system with non linear thermal diffusion. Nonlinear Anal. Theory Methods Appl. 30(6), 3255–3263 (1997)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Drazin P.G., Reid W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)MATHGoogle Scholar
  5. 5.
    Climent-Ezquerra B., Guillén-González F., Rojas-Medar M.A.: Time-periodic solutions for a generalized Boussinesq model with Neumann boundary conditions for temperature. Proc. R. Soc. A 463, 2153–2164 (2007)CrossRefMATHADSGoogle Scholar
  6. 6.
    Feireisl E.: Dynamics of Viscous Incompressible Fluids. Oxford University Press, Oxford (2004)Google Scholar
  7. 7.
    Feireisl, E., Málek, J.: On the Navier–Stokes equations with temperature dependent transport coefficients. Differ. Equ. Nonlinear Mech. Art. ID 90616, pp. 14 (2006) (electronic)Google Scholar
  8. 8.
    Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I. Springer, New York (1994)CrossRefGoogle Scholar
  9. 9.
    Guillén-González F., Damázio P., Rojas-Medar M.A.: Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion. J. Math. Anal. Appl. 326, 468–487 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hishida T.: Existence and regularizing properties of solutions for the nonstationary convection problem. Funkcialy Ekvaciy 34, 449–474 (1991)MATHMathSciNetGoogle Scholar
  11. 11.
    Kagei Y.: On weak solutions of nonstationary Boussinesq equations. Differ. Integr. Equ. 6(3), 587–611 (1993)MATHMathSciNetGoogle Scholar
  12. 12.
    Kagei Y., von Wahl W.: Stability of higher norms in terms of energy-stability for the Boussinesq equations: remarks on the asymptotic behaviour of convection-roll-type solutions. Differ. Integr. Equ. 7(3–4), 921–948 (1994)MATHGoogle Scholar
  13. 13.
    Joseph D.D.: Stability of Fluid Motion. Springer, Berlin (1976)Google Scholar
  14. 14.
    Lorca S.A., Boldrini J.L.: Stationary solutions for generalized Boussinesq models. J. Difer. Equ. 124, 389–406 (1996)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lorca S.A., Boldrini J.L.: The initial value problem for a generalized Boussinesq model. Nonlinear Anal. 36, 457–480 (1999)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Málek J., Ruzicka M., Thäter G.: Fractal dimension, attractors, and the Boussinesq approximation in three dimensions, Mathematical problems for Navier–Stokes equations (Centro, 1993). Acta Appl. Math. 37(1–2), 83–97 (1994)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Moretti A.C., Rojas-Medar M.A., Rojas Medar M.D.: Reproductive weak solutions for generalized Boussinesq models in exterior domains. Mat. Contemp. 23, 119–137 (2002)MATHMathSciNetGoogle Scholar
  18. 18.
    Moretti A.C., Rojas-Medar M.A., Drina Rojas-Medar M.: The equations of a viscous incompressible chemically active fluid: existence and uniqueness of strong solutions in an unbounded domain. Comput. Math. Appl. 44, 287–299 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Morimoto H.: Nonstationary Boussinesq equations. J. Fac. Sci., Univ Tokyo, Sect., IA Math. 39, 61–75 (1992)MATHMathSciNetGoogle Scholar
  20. 20.
    Notte-Cuello E.A., Rojas-Medar M.A.: Stationary solutions for generalized Boussinesq models in exterior domains. Electron. J. Differ. Equ., 1998(22), 1–9 (1998)MathSciNetGoogle Scholar
  21. 21.
    Rajagopal K.R., Ruzicka M., Srinivasa A.R.: On the Oberbeck–Boussinesq approximation. Math. Models Methods Appl. Sci. 6(8), 1157–1167 (1996)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rojas-Medar M.A., Lorca S.A.: The equation of a viscous incompressible chemical active fluid. I. uniqueness and existence of the local solutions. Rev. Mat. Apl. 16, 57–80 (1995)MATHMathSciNetGoogle Scholar
  23. 23.
    Rojas-Medar M.A., Lorca S.A.: Global strong solution of the equations for the motion of a chemical active fluid. Mat. Contemp. 8, 319–335 (1995)MATHMathSciNetGoogle Scholar
  24. 24.
    Rojas-Medar M.A., Lorca S.A.: An error estimate uniform in time for spectral Galerkin approximations for the equations for the motion of chemical active fluid. Rev. Univ. Complutense de Madrid. 18, 431–458 (1995)MathSciNetGoogle Scholar
  25. 25.
    Sohr H.: The Navier–Stokes Equatios. An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)Google Scholar
  26. 26.
    von Tippelkirch H.: Über Konvektionszeller insbesondere in flüssigen Schefel. Beiträge Phys. Atmos. 20, 37–54 (1956)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • José Luiz Boldrini
    • 1
  • Blanca Climent-Ezquerra
    • 2
  • María Drina Rojas-Medar
    • 3
  • Marko A. Rojas-Medar
    • 4
  1. 1.IMECC-UNICAMPCampinasBrazil
  2. 2.Dpto. de Ecuaciones Diferenciales y Análisis Numérico Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  3. 3.Dpto. de MatemáticaUniversidad de AntofagastaAntofagastaChile
  4. 4.Dpto. de Ciencias Básicas, Facultad de CienciasUniversidad del Bío-BíoChillánChile

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