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Analysis of an iterative method for variable density incompressible fluids

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Abstract

We derive error bounds for an iterative method used by Okamoto to prove the existence of strong solutions for the equations of nonhomogeneous incompressible fluids.

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References

  1. Amrouche C., Girault V.: On the existence and regularity of the solutions of Stokes problem in arbitrary dimension. Proc. Japan Acad. Ser. A Math. Sci. 67(5), 171–175 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Studies in Mathematics and its Applications, vol. 22, 309 p. North-Holland, Amsterdam (1990)

  3. Boldrini J.L., Rojas-Medar M.A.: Global solutions to the equations for the motion of stratified incompressible fluids. Mat. Contemp. 3, 1–8 (1992)

    MATH  MathSciNet  Google Scholar 

  4. Boldrini J.L., Rojas-Medar M.A.: Global strong solutions of the equations for the motion of nonhomogeneous incompressible fluids. In: Conca, C., Gatica, G. (eds) Numerical Methods in Mechanics, Pitman Res. Notes Math. Series 371, pp. 35–45. Longman, Harlow (1997)

    Google Scholar 

  5. Boldrini J.L., Rojas-Medar M.A.: An error estimate uniform in time for spectral semi-Galerkin approximations of the nonhomogeneous Navier-Stokes equations. Numer. Funct. Anal. Optim. 15(7–8), 755–778 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Braz e Silva, P., Rojas-Medar, M.A.: Error bounds for semi-Galerkin approximations of nonhomogeneous incompressible fluids. J. Math. Fluid Mech. doi:10.1007/s00021-007-0255-9

  7. Cattabriga L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)

    MATH  MathSciNet  Google Scholar 

  8. Conca C., Gormaz R., Ortega-Torres E., Rojas-Medar M.A.: The equations of non-homogeneous asymmetric fluids: an iterative approach. Math. Meth. Appl. Sci. 25(15), 1251–1280 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Frolova E., Padula M.: Free boundary problem for a layer of inhomogeneous fluid. Eur. J. Mech. B Fluids 23(4), 665–679 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fujita H., Kato T.: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16(5), 269–315 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  11. Guillén-Gonzalez 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(1), 468–487 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Heywood J.G.: An error estimate uniform in time for spectral semi-Galerkin approximations of the Navier-Stokes problem. Pacific J. Math. 98(2), 333–345 (1982)

    MATH  MathSciNet  Google Scholar 

  13. Heywood J.G., Rannacher R.: Finite element approximation of the nonstationary Navier-Stokes problem I: regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Kim J.U.: Weak solutions of an initial-boundary value problems for an incompressible viscous fluid with nonnegative density. SIAM J. Math. Anal. 18(1), 89–96 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ladyzhenskaya, O.A., Solonnikov, V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. In: Boundary value problems of mathematical physics, and related questions of the theory of functions 52, Zap. Naučn Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), pp. 52–109 (1975)

  17. Okamoto H.: On the equation of nonstationary stratified fluid motion: uniqueness and existence of the solutions. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30(3), 615–643 (1984)

    MATH  MathSciNet  Google Scholar 

  18. Padula M.: An existence theorem for nonhomogeneous incompressible fluids. Rend. Circ. Mat. Palermo 2(Suppl. 31), 119–124 (1982)

    Article  MathSciNet  Google Scholar 

  19. Salvi R.: Error estimates for the spectral Galerkin approximations of the solutions of Navier-Stokes type equations. Glasgow Math. J. 31(2), 199–211 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Salvi R.: The equations of viscous incompressible nonhomogeneous fluid: on the existence and regularity. J. Austral. Math. Soc. Ser. B 33(1), 94–110 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  21. Simon J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Temam, R.: Navier-Stokes equations: theory and numerical analysis, 3rd edn. In: Studies in Mathematics and its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1984)

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Authors and Affiliations

  1. Depto. de Matemáticas, Univ. de Antofagasta, Casilla 170, Antofagasta, Chile

    Elva Ortega-Torres

  2. Depto. de Matemática, Univ. Federal de Pernambuco, Recife, PE, 50740-540, Brazil

    Pablo Braz e Silva

  3. Dpto. de Matemática Aplicada, Univ. Estadual de Campinas, Campinas, SP, 13083-970, Brazil

    Marko Rojas-Medar

Authors
  1. Elva Ortega-Torres
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  2. Pablo Braz e Silva
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  3. Marko Rojas-Medar
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Corresponding author

Correspondence to Elva Ortega-Torres.

Additional information

The authors Ortega-Torres and Rojas-Medar are partially supported by FONDECYT-Chile grant No. 1040205 and No. 7040187. The authors Braz e Silva and Rojas-Medar are partially supported by CAPES/MECD-DGU Brazil/Spain grant No. 117/06.

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Ortega-Torres, E., Braz e Silva, P. & Rojas-Medar, M. Analysis of an iterative method for variable density incompressible fluids. Ann. Univ. Ferrara 55, 129–151 (2009). https://doi.org/10.1007/s11565-009-0060-x

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  • DOI: https://doi.org/10.1007/s11565-009-0060-x

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