Advertisement

Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1917–1981 | Cite as

Existence of Bifurcating Quasipatterns in Steady Bénard–Rayleigh Convection

  • Boele Braaksma
  • Gérard IoossEmail author
Article
  • 21 Downloads

Abstract

Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Bénard–Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle \({\pi/q}\). There is a small divisor problem for \({q \geqq 4}\).

Using the results of Berti–Bolle–Procesi in 2010, we adapt it to a Navier–Stokes system ruling the Bénard–Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift–Hohenberg PDE model. For using the Nash–Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point.

There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of L(N)L(N)* which depends mainly quadratically on the main parameter. A careful study of the “bad set”of parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti et al. for solving ”the range equation”. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional “bifurcation equation.

For any \({q \geqq 4}\) , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors warmly thank Michela Procesi and also Laurent Stolovitch, Philippe Bolle, Maximilliano Berti, and Nicolas Burq for the interactions they had about this work, especially during the Winter schools at St Etienne de Tinée in February 2016 and 2017.We warmly thank the referee who was extremely efficient in his (her) criticisms, detecting several mistakes in the original version; this forced us to considerably clarify our presentation.

References

  1. 1.
    Berti M., Bolle P.: Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions. Arch. Rat. Mech. Anal. 195(2), 609–642 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berti M., Bolle P., Procesi M.: An abstract Nash–Moser theorem with parameters and applications to PDEs. Ann. Inst. Poincaré Anal. Non Linéaire, 27(1), 377–399 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Braaksma B., Iooss G., Stolovitch L.: Existence proof of quasipatterns solutions of the Swift–Hohenberg equation. Commun. Math. Phys. 353(1), 37–67 (2017)  https://doi.org/10.1007/s00220-017-2878-x ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal., 5(4), 629–639 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford (1961)zbMATHGoogle Scholar
  6. 6.
    Christiansen B., Alstrom P., Levinsen M.T.: Ordered capillary-wave states Quasi crystals, hexagons, and radial waves. Phys. Rev. Lett., 68(14), 2157–2160 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    Craig W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles, Vol. 9 of Panoramas et Synthèses. Société Mathématiques de France, Paris (2000)Google Scholar
  8. 8.
    Edwards W.S., Fauve S.: Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech., 278, 123–148 (1994)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Görtler H., Kirchgässner K., Sorger P.: Branching solutions of the Bénard problem. Problems of Hydrodynamics and continuum mechanics. NAUKA, Moscow (1969) 133–149zbMATHGoogle Scholar
  10. 10.
    Iooss, G.: Quasipatterns in steady Bénard–Rayleigh convection. Izvestiya Vuzov Severo–Kavkazskii Region, Special Issue “Actual problems of mathematical hydrodynamics”, Natural Science, pp. 92–105. Volume in honor of 75th anniversary of the birth of V.Yudovich 2009Google Scholar
  11. 11.
    Iooss G., Rucklidge A.M.: On the existence of quasipattern solutions of the Swift–Hohenberg equation. J. Nonlinear Sci. 20, 361–394 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Joseph D.D.: Stability of Fluid Motions. I and II Springer Tracts in natural Philosophy. Vols. 27 and 28. Springer-Verlag, Berlin (1976)Google Scholar
  13. 13.
    Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer- Verlag, Berlin (1995) Reprint of the 1980 editionCrossRefzbMATHGoogle Scholar
  14. 14.
    Kirchgässner K., Kielhofer H.J.: Stability and bifurcation in fluid mechanics. Rocky Mt. J. Math. 3(2), 275–318 (1973)CrossRefzbMATHGoogle Scholar
  15. 15.
    Koschmieder E.L.: Bénard Cells and Taylor Vortices. Cambridge Monographs on Mechanics and Applied Maths. Cambridge University Press, Cambridge (1993) Google Scholar
  16. 16.
    Lions J.L., Magenes E.: Problèmes aux limites non homogènes., Vol. 1 Travaux et Recherches mathématiques. mathématiques, Dunod (1968)zbMATHGoogle Scholar
  17. 17.
    Rabinowitz P.H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rat. Mech. Anal. 29(1), 32–57 (1968)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rogers J.L., Pesch W., Brausch O., Schatz M.F.: Complex-ordered patterns in shaken convection. Phys. Rev. E, 71(6), 066214 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Rucklidge A.M., Rucklidge W.J.: Convergence properties of the 8, 10 and 12 mode representations of quasi-patterns. Physica D, 178, 62–82 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rucklidge, A.M., Silber, M.: Design of parametrically forced patterns and quasipatterns. SIAM J. Appl. Dyn. Syst. 2009Google Scholar
  21. 21.
    Ukhovskii M.R., Yudovich V.I.: On the equations of steady State convection. J. Appl. Math. Mech. 27(2), 432–440 (1963)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Velte W.: Konvexität der Eigenkurven beim Bénardschen Problem. Z.A.M.P. 20(5), 636–641 (1969)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Volmar U.E., Muller H.W.: Quasiperiodic patterns in Rayleigh–Bénard convection under gravity modulation. Phys. Rev. E, 56(5), 5423–5430 (1997)ADSCrossRefGoogle Scholar
  24. 24.
    Washington L.C.: Introduction to Cyclotomic Fields, Vol. 83,Graduate Texts in Mathematics. Springer-Verlag., New york (1997) 2nd ed. 1997CrossRefGoogle Scholar
  25. 25.
    Yudovich V.I.: On the origin of convection. J. Appl. Math. Mech. 30(6), 1193–1199 (1966)CrossRefGoogle Scholar
  26. 26.
    Yudovich V.I.: Free convection and bifurcation. J. Appl. Math. Mech. 31(1), 103–114 (1967)CrossRefzbMATHGoogle Scholar
  27. 27.
    Yudovich V.I.: Stability of convection flows. J. Appl. Math. Mech. 31(2), 294–303 (1967)CrossRefzbMATHGoogle Scholar
  28. 28.
    Yudovich, V.I.: The LinearizationMethod in Hydrodynamical Stability Theory. Transl. Math. Monographs, Vol. 74, AMS 1989Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Johann Bernoulli InstituteUniversity of GroningenGroningenThe Netherlands
  2. 2.Université Côte d’Azur, CNRS, LJAD, Institut Universitaire de FranceNice Cedex 02France

Personalised recommendations