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Error Estimates for Spectral Semi-Galerkin Approximations of Incompressible Asymmetric Fluids with Variable Density

  • Felipe W. CruzEmail author
  • Pablo Braz e Silva
Article

Abstract

We consider spectral semi-Galerkin approximations for global strong solutions of the equations for variable density asymmetric incompressible fluids in a bounded domain \(\Omega \) of \(\mathbb {R}^3\). We prove an optimal uniform in time error estimate in the \({\varvec{H}}^{1}\) norm for approximations of both the linear and angular velocity of particles of the fluid. We also derive an error bound for approximations of the density in some Lebesgue spaces \(L^{r}(\Omega )\).

Keywords

Semi-Galerkin approximations Error estimate uniform in time Asymmetric fluids 

Mathematics Subject Classification

65M60 76M22 65M15 35Q30 35Q35 

Notes

Acknowledgements

P. Braz e Silva was partially supported by CNPq, Brazil. Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant No. 309491/2015-0).

Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

References

  1. 1.
    Amann, H.: Ordinary Differential Equations. An Introduction to Nonlinear Analysis. Translated from the German by Gerhard Metzen. De Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter & Co., Berlin (1990)Google Scholar
  2. 2.
    Amrouche, C., Girault, V.: On the existence and regularity of the solution of Stokes problem in arbitrary dimension. Proc. Jpn. Acad. Ser. A Math. Sci. 67(5), 171–175 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value problems in Mechanics of Nonhomogeneous Fluids. Translated from the Russian. Studies in Mathematics and its Applications, vol. 22. North-Holland Publishing Co., Amsterdam (1990)Google Scholar
  4. 4.
    Boldrini, J.L., Notte-Cuello, E., Poblete-Cantellano, M., Friz, L., Rojas-Medar, M.A.: Pointwise error estimate for spectral Galerkin approximations of micropolar equations. Numer. Funct. Anal. Optim. 37(3), 304–323 (2016)MathSciNetCrossRefGoogle Scholar
  5. 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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Boldrini, J.L., Rojas-Medar, M.A.: Global solutions to the equations for the motion of stratified incompressible fluids Second. Workshop on Partial Differential Equations (Rio de Janeiro, 1991). Mat. Contemp. 3, 1–8 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    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.N. (eds.), Numerical Methods in Mechanics, Pitman Research Notes in Mathematics Series, vol. 371, pp. 35–45. Longman, Harlow (1997)Google Scholar
  8. 8.
    Boldrini, J.L., Rojas-Medar, M.A., Fernández-Cara, E.: Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids. J. Math. Pures Appl. (9) 82(11), 1499–1525 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Braz e Silva, P., Cruz, F.W., Rojas-Medar, M.A.: Semi-strong and strong solutions for variable density asymmetric fluids in unbounded domains. Math. Methods Appl. Sci. 40(3), 757–774 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Braz e Silva, P., Cruz, F.W., Rojas-Medar, M.A.: Vanishing viscosity for non-homogeneous asymmetric fluids in \(\mathbb{R}^3\): the \(L^2\) case. J. Math. Anal. Appl. 420(1), 207–221 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Braz e Silva, P., Cruz, F.W., Rojas-Medar, M.A., Santos, E.G.: Weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum. J. Math. Anal. Appl. 473(1), 567–586 (2019)CrossRefGoogle Scholar
  12. 12.
    Braz e Silva, P., Fernàndez-Cara, E., Rojas-Medar, M.A.: Vanishing viscosity for non-homogeneous asymmetric fluids in \(\mathbb{R}^3\). J. Math. Anal. Appl. 332(2), 833–845 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Braz e Silva, P., Rojas-Medar, M.A.: Error bounds for semi-Galerkin approximations of nonhomogeneous incompressible fluids. J. Math. Fluid Mech. 11(2), 186–207 (2009)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Braz e Silva, P., Santos, E.G.: Global weak solutions for asymmetric incompressible fluids with variable density. C. R. Math. Acad. Sci. Paris 346(9–10), 575–578 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Braz e Silva, P., Santos, E.G.: Global weak solutions for variable density asymmetric incompressible fluids. J. Math. Anal. Appl. 387(2), 953–969 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)MathSciNetGoogle Scholar
  18. 18.
    Heywood, J.G.: An error estimate uniform in time for spectral Galerkin approximations of the Navier–Stokes problem. Pac. J. Math. 98(2), 333–345 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    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)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kažihov, A.V.: Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid. Dokl. Akad. Nauk SSSR 216, 1008–1010 (1974)ADSMathSciNetGoogle Scholar
  21. 21.
    Kim, J.U.: Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density. SIAM J. Math. Anal. 18(1), 89–96 (1987)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ladyženskaja, O.A., Solonnikov, V.A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. Boundary value problems of mathematical physics, and related questions of the theory of functions, 8. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52(52–109), 218–219 (1975)Google Scholar
  23. 23.
    Lions, P.-L.: Mathematical Topics in Fluid Mechanics vol. 1. Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford Science Publications. The Clarendon Press/Oxford University Press, New York (1996)Google Scholar
  24. 24.
    Lions, J.-L.: On some problems connected with Navier–Stokes equations. In: Nonlinear Evolution Equations (Proceedings of Symposium, University of Wisconsin-Madison, 1977), Publication of the Mathematics Research Center Wisconsin, vol. 40, , pp. 59–84. Academic Press, New York (1978)Google Scholar
  25. 25.
    Lions, J.-L.: On some questions in boundary value problems of mathematical physics. Contemporary developments in continuum mechanics and partial differential equations (Proceedings of the International Symposium, Institute of Mathematics Universidade Federal Rio de Janeiro, Rio de Janeiro, 1977, North-Holland Mathematics Studies, vol. 30, ), pp. 284–346. North-Holland, Amsterdam (1978)Google Scholar
  26. 26.
    Łukaszewicz, G.: Micropolar Fluids Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston Inc., Boston (1999)zbMATHGoogle Scholar
  27. 27.
    Łukaszewicz, G.: On nonstationary flows of incompressible asymmetric fluids. Math. Methods Appl. Sci. 13(3), 219–232 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    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)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ortega-Torres, E.E., Rojas-Medar, M.A., Cabrales, R.C.: A uniform error estimate in time for spectral Galerkin approximations of the magneto-micropolar fluid equations. Numer. Methods Partial Differ. Equ. 28(2), 689–706 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ortega-Torres, E.E., Villamizar-Roa, E.J., Rojas-Medar, M.A.: Micropolar fluids with vanishing viscosity. Abstr. Appl. Anal. 2010, 1–18 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rautmann, R.: On the convergence rate of nonstationary Navier–Stokes approximations. In: Approximation Methods for Navier–Stokes Problems (Proceedings of the Symposium, University of Paderborn, Paderborn, 1979), Lecture Notes in Mathematics, vol. 771, , pp. 425–449. Springer, Berlin (1980)Google Scholar
  32. 32.
    Rojas-Medar, M.A.: Magneto-micropolar fluid motion: on the convergence rate of the spectral Galerkin approximations. Z. Angew. Math. Mech. 77(10), 723–732 (1997)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Rojas-Medar, M.A., Boldrini, J.L.: Spectral Galerkin approximations for the Navier–Stokes equations: uniform in time error estimates. Rev. Mat. Apl. 14(2), 63–74 (1993)MathSciNetzbMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1977)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de PernambucoRecifeBrazil

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