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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 2, pp 517–529 | Cite as

Estimates of Time-Periodic Fundamental Solutions to the Linearized Navier–Stokes Equations

  • Thomas Eiter
  • Mads Kyed
Article

Abstract

Fundamental solutions to the time-periodic Stokes and Oseen linearizations of the Navier–Stokes equations in dimension \(n\ge 2\) are investigated. Integrability properties and pointwise estimates are established.

Keywords

Stokes Oseen Navier–Stokes Time-periodic Fundamental solution 

Mathematics Subject Classification

Primary 35Q30 35B10 35A08 35E05 76D07 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th printing, with corrections. Wiley, New York (1972)Google Scholar
  2. 2.
    Bruhat, F.: Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes \(p\)-adiques. Bull. Soc. Math. Fr. 89, 43–75 (1961)CrossRefMATHGoogle Scholar
  3. 3.
    Edwards, R., Gaudry, G.: Littlewood-Paley and Multiplier Theory. Springer, Berlin (1977)CrossRefMATHGoogle Scholar
  4. 4.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011)MATHGoogle Scholar
  5. 5.
    Galdi, G.P., Kyed, M.: Time-periodic solutions to the Navier–Stokes equations in the three-dimensional whole space with a non-zero drift term: asymptotic profile at spatial infinity. arXiv:1610.00677 (2016)
  6. 6.
    Geissert, M., Hieber, M., Nguyen, T.H.: A general approach to time periodic incompressible viscous fluid flow problems. Arch. Ration. Mech. Anal. 220(3), 1095–1118 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Grafakos, L.: Classical Fourier analysis, 2nd edn. Springer, New York (2008)MATHGoogle Scholar
  8. 8.
    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Kozono, H., Nakao, M.: Periodic solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. (2) 48(1), 33–50 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kyed, M.: Maximal regularity of the time-periodic linearized Navier–Stokes system. J. Math. Fluid Mech. 16(3), 523–538 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kyed, M.: A fundamental solution to the time-periodic Stokes equations. J. Math. Anal. Appl. 437(1), 708719 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Prodi, G.: Qualche risultato riguardo alle equazioni di Navier–Stokes nel caso bidimensionale. Rend. Sem. Mat. Univ. Padova 30, 1–15 (1960)MathSciNetMATHGoogle Scholar
  13. 13.
    Prouse, G.: Soluzioni periodiche dell’equazione di Navier–Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 35, 443–447 (1963)MathSciNetMATHGoogle Scholar
  14. 14.
    Serrin, J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Stakgold, I.: Boundary value problems of mathematical physics. Vol. I and II. Reprint of the 1967/68 originals. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000)Google Scholar
  16. 16.
    Yudovich, V.: Periodic motions of a viscous incompressible fluid. Sov. Math. Dokl. 1, 168–172 (1960)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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