Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1035–1058 | Cite as

Existence of Time Periodic Solution to Some Double-Diffusive Convection System in the Whole Space Domain

  • Mitsuharu Ôtani
  • Shun Uchida
Article

Abstract

This paper is concerned with the existence of time periodic solutions to some system which describes double-diffusive convection phenomena in the whole space \({\mathbb {R}} ^N \) with \(N = 3\) and 4. In previous results for periodic problems of parabolic type equations with non-monotone perturbation terms, it seems that either of the smallness of given data or the boundedness of space domain is essential. In spite of the presence of non-monotone terms, the solvability of our problem in the whole space is shown for large external forces via the convergence of solutions to approximate equations in bounded domains.

Keywords

Time periodic problem Whole space domain Large data Double-diffusive convection Brinkman–Forchheimer equation 

Mathematics Subject Classification

Primary 35B10 Secondary 35K40 35Q35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to thank the editor and the referees for carefully reading the manuscript and for giving constructive comments which substantially helped improving the quality of this paper. The first author was partially supported by the Grant-in-Aid for Scientific Research [Grant Number 15K13451], the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. The second author was supported by the Grant-in-Aid for JSPS Fellows [Grant Number 26\(\cdot \)5316], Japan Society for the Promotion of Science (JSPS). Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

References

  1. 1.
    Bénilan, P., Brézis, H.: Solutions faibles d’équations d’évolution dans les espaces de Hilbert. Ann. Inst. Fourier (Grenoble) 22(2), 311–329 (1972)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brandt, A., Fernando, H.J.S. (eds.): Double-diffusive convection. Geophysical Monograph Series, vol. 94. American Geophysical Union, Washington, D.C. (1995)Google Scholar
  3. 3.
    Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland Publishing Co., Amsterdam (1973)MATHGoogle Scholar
  4. 4.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems. Springer, Berlin (2011)MATHGoogle Scholar
  5. 5.
    Galdi, G.P., Kyed, M.: Time-periodic solutions to the Navier-Stokes equations. In: Giga, Y., Novotny, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 1–70. Springer, Berlin (2016)Google Scholar
  6. 6.
    Inoue, H., Ôtani, M.: Periodic problems for heat convection equations in noncylindrical domains. Funkcial. Ekvac. 40(1), 19–39 (1997)MathSciNetMATHGoogle Scholar
  7. 7.
    Kozono, H., Nakao, M.: Periodic solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. 48, 33–50 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maremonti, P.: Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space. Nonlinearity 4, 503–529 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nagai, T.: Periodic solutions for certain time-dependent parabolic variational inequalities. Hiroshima Math. J. 5, 537–549 (1975)MathSciNetMATHGoogle Scholar
  10. 10.
    Nield, D.A., Bejan, A.: Convection in Porous Medium, 4th edn. Springer, Berlin (2012)MATHGoogle Scholar
  11. 11.
    Ôtani, M.: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems. J. Differ. Equ. 54, 248–273 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ôtani, M.: \(L ^{\infty }\)-energy method, basic tools and usage. In: Staicu, V. (ed.) Progress in Nonlinear Differential Equations and Their Applications, Progress in Nonlinear Differential Equations and Their Applications, vol. 75, pp. 357–376. Birkhäuser Verlag, Basel (2008)Google Scholar
  13. 13.
    Ôtani, M., Uchida, S.: The existence of periodic solutions of some double-diffusive convection system based on Brinkman–Forchheimer equations. Adv. Math. Sci. Appl. 23(1), 77–92 (2013)MathSciNetMATHGoogle Scholar
  14. 14.
    Ôtani, M., Uchida, S.: Global solvability for double-diffusive convection system based on Brinkman-Forchheimer equation in general domains. Osaka J. Math. 53(3), 855–872 (2016)MathSciNetMATHGoogle Scholar
  15. 15.
    Radko, T.: Double-diffusive Convection. Cambridge University Press, Cambridge (2013)CrossRefMATHGoogle Scholar
  16. 16.
    Sohr, H.: The Navier–Stokes Equations, An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)MATHGoogle Scholar
  17. 17.
    Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis, Third revised edition. North Holland Publishing Co., Amsterdam (1984)Google Scholar
  18. 18.
    Villamizar-Roa, E.J., Rodríguez-Bellido, M.A., Rojas-Medar, M.A.: Periodic solution in unbounded domains for the Boussinesq system. Acta Math. Sin. (Engl. Ser.) 26(5), 837–862 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yamada, Y.: Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries. Nagoya Math. J. 70, 111–123 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied Physics, School of Advanced Science and EngineeringWaseda UniversityTokyoJapan

Personalised recommendations