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

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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.

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References

  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)

  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)

    Article  MATH  Google Scholar 

  3. Edwards, R., Gaudry, G.: Littlewood-Paley and Multiplier Theory. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  4. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011)

    MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grafakos, L.: Classical Fourier analysis, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  8. Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York (2009)

    Book  MATH  Google Scholar 

  9. Kozono, H., Nakao, M.: Periodic solutions of the Navier–Stokes equations in unbounded domains. Tohoku Math. J. (2) 48(1), 33–50 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kyed, M.: Maximal regularity of the time-periodic linearized Navier–Stokes system. J. Math. Fluid Mech. 16(3), 523–538 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kyed, M.: A fundamental solution to the time-periodic Stokes equations. J. Math. Anal. Appl. 437(1), 708719 (2016)

    Article  MathSciNet  Google Scholar 

  12. Prodi, G.: Qualche risultato riguardo alle equazioni di Navier–Stokes nel caso bidimensionale. Rend. Sem. Mat. Univ. Padova 30, 1–15 (1960)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  14. Serrin, J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. Yudovich, V.: Periodic motions of a viscous incompressible fluid. Sov. Math. Dokl. 1, 168–172 (1960)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Thomas Eiter & Mads Kyed

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  1. Thomas Eiter
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  2. Mads Kyed
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Correspondence to Mads Kyed.

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The authors declare that they have no conflict of interest.

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Communicated by Y. Giga.

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Eiter, T., Kyed, M. Estimates of Time-Periodic Fundamental Solutions to the Linearized Navier–Stokes Equations. J. Math. Fluid Mech. 20, 517–529 (2018). https://doi.org/10.1007/s00021-017-0332-7

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  • DOI: https://doi.org/10.1007/s00021-017-0332-7

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