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Asymptotics and Lower Bound for the Lifespan of Solutions to the Primitive Equations

  • Frédéric Charve
Article
  • 34 Downloads

Abstract

In a previous work we obtained a large lower bound for the lifespan of the solutions to the Primitive Equations, and proved convergence to the 3D quasi-geostrophic system for general and ill-prepared blowing-up data, when the kinematic viscosity\(\nu \)is equal to the heat diffusivity\(\nu '\), turning the diffusion operator \(\varGamma \) into the classical Laplacian.

Obtaining the same results in the general case is much more difficult as it involves a homogeneous non-local non-radial diffusion operator \(\varGamma \) whose semi-group and singular integral form kernels present sign changes. Every classical result related to non-local operators, or to Navier-Stokes system then becomes more involved here and the key ingredient will be new transport-diffusion estimates obtained in a companion paper and a precise use of the quasi-geostrophic decomposition.

Keywords

Navier-Stokes system Besov spaces Littlewood-Paley decomposition Primitive equations 3D quasi-geostrophic system Strichartz estimates Non-local operators Boussinesq system 

Notes

Acknowledgements

The author wishes to thank R. Danchin, I. Gallagher, and T. Hmidi for useful discussions.

References

  1. 1.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids. Eur. J. Mech. B, Fluids 15, 291–300 (1996) zbMATHGoogle Scholar
  2. 2.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babin, A., Mahalov, A., Nicolaenko, B.: Strongly stratified limit of 3D primitive equations in an infinite layer. In: Advances in Wave Interaction and Turbulence, South Hadley, MA, 2000. Contemp. Math., vol. 283. Am. Math. Soc., Providence (2001) Google Scholar
  4. 4.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011) zbMATHGoogle Scholar
  5. 5.
    Beale, T., Bourgeois, A.: Validity of the quasi-geostrophic model for large scale flow in the atmosphere and ocean. SIAM J. Math. Anal. 25, 1023–1068 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Éc. Norm. Supér. 14, 209–246 (1981) CrossRefzbMATHGoogle Scholar
  7. 7.
    Bougeault, P., Sadourny, R.: Dynamique de l’Atmosphère et de l’Océan. Editions de l’Ecole Polytechnique, Paris (2001) Google Scholar
  8. 8.
    Charve, F.: Global well-posedness and asymptotics for a geophysical fluid system. Commun. Partial Differ. Equ. 29 (11 & 12), 1919–1940 (2004) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Charve, F.: Convergence of weak solutions for the primitive system of the quasigeostrophic equations. Asymptot. Anal. 42, 173–209 (2005) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Charve, F.: Asymptotics and vortex patches for the quasigeostrophic approximation. J. Math. Pures Appl. 85, 493–539 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Charve, F.: Global well-posedness for the primitive equations with less regular initial data. Ann. Fac. Sci. Toulouse 17(2), 221–238 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Charve, F., Ngo, V-S.: Asymptotics for the primitive equations with small anisotropic viscosity. Rev. Mat. Iberoam. 27(1), 1–38 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Charve, F., Danchin, R.: A global existence result for the compressible Navier-Stokes equations in the critical Lp framework. Arch. Ration. Mech. Anal. 198(1), 233–271 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Charve, F., Haspot, B.: On a Lagrangian method for the convergence from a non-local to a local Korteweg capillary fluid model. J. Funct. Anal. 265(7), 1264–1323 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Charve, F.: Convergence of a low order non-local Navier-Stokes-Korteweg system: the order-parameter model. Asymptot. Anal. 100(3–4), 153–191 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Charve, F.: A priori estimates for the 3D quasi-geostrophic system. J. Math. Anal. Appl. 444(2), 911–946 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chemin, J.-Y.: Fluides parfaits incompressibles. Astérisque 230 (1995) Google Scholar
  18. 18.
    Chemin, J.-Y.: A propos d’un problème de pénalisation de type antisymétrique. J. Math. Pures Appl. 76, 739–755 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Anisotropy and dispersion in rotating fluids. In: Nonlinear Partial Differential Equations and Their Application, Collège de France Seminar. Studies in Mathematics and its Applications, vol. 31, pp. 171–191 (2002) Google Scholar
  20. 20.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity, Special issue for R. Temam’s 60th birthday. Modél. Math. Anal. Numér. 34(2), 315–335 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Ekman boundary layers in rotating fluids. ESAIM Control Optim. Calc. Var. 8, 441–466 (2002). A tribute to J.-L. Lions MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier-Stokes Equations. Oxford University Press, London (2006) zbMATHGoogle Scholar
  23. 23.
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Cushman-Roisin, B.: Introduction to Geophysical Fluid Dynamics. Prentice Hall, New York (1994) zbMATHGoogle Scholar
  25. 25.
    Danchin, R.: Poches de tourbillon visqueuses. J. Math. Pures Appl. 76(7), 609–647 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Desjardins, B., Grenier, E.: Derivation of the quasigeostrophic potential vorticity equations. Adv. Differ. Equ. 3(5), 715–752 (1998) zbMATHGoogle Scholar
  27. 27.
    Dutrifoy, A., Hmidi, T.: The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data. Commun. Pure Appl. Math. 57(9), 1159–1177 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Dutrifoy, A.: Slow convergence to vortex patches in quasigeostrophic balance. Arch. Ration. Mech. Anal. 171(3), 417–449 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dutrifoy, A.: Examples of dispersive effects in non-viscous rotating fluids. J. Math. Pures Appl. 84(3), 331–356 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Embid, P., Majda, A.: Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial Differ. Equ. 21, 619–658 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Embid, P., Majda, A.: Averaging over fast gravity waves for geophysical flows with unbalanced initial data. Theor. Comput. Fluid Dyn. 11, 155–169 (1998) CrossRefzbMATHGoogle Scholar
  32. 32.
    Gallagher, I.: Applications of Schochet’s methods to parabolic equation. J. Math. Pures Appl. 77, 989–1054 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gamblin, P., Saint Raymond, X.: On three dimensional vortex patches. Bull. Soc. Math. Fr. 123(3), 375–424 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Greenspan, H.P.: The Theory of Rotating Fluids. Cambridge University Press, Cambridge (1968) zbMATHGoogle Scholar
  35. 35.
    Hmidi, T.: Régularité hölderienne des poches de tourbillon visqueuses. J. Math. Pures Appl. 84(11), 1455–1495 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hmidi, T., Keraani, S.: On the global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces. Adv. Math. 214(2), 618–638 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hmidi, T., Abidi, H.: On the global well-posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal. 40(1), 167–185 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hmidi, T., Zerguine, M.: On the global well-posedness of the Euler-Boussinesq system with fractional dissipation. Physica D 239(15), 1387–1401 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Iftimie, D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Rev. Mat. Iberoam. 15, 1–36 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Iftimie, D.: The approximation of the quasigeostrophic system with the primitive systems. Asymptot. Anal. 21(2), 89–97 (1999) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Konieczny, P., Yoneda, T.: On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations. J. Differ. Equ. 250, 3859–3873 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lions, J.-L., Temam, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237–288 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lions, J.-L., Temam, R., Wang, S.: Geostrophic asymptotics of the primitive equations of the atmosphere. Topol. Methods Nonlinear Anal. 4, 1–35 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ngo, V.-S.: Rotating fluids with small viscosity. Int. Math. Res. Not. 2009(10), 1860–1890 (2009) MathSciNetzbMATHGoogle Scholar
  45. 45.
    Paicu, M.: Étude asymptotique pour les fluides anisotropes en rotation rapide dans le cas périodique. J. Math. Pures Appl. 83(2), 163–242 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Paicu, M.: Équation périodique de Navier-Stokes sans viscosité dans une direction. Commun. Partial Differ. Equ. 30(7–9), 1107–1140 (2005) CrossRefzbMATHGoogle Scholar
  47. 47.
    Paicu, M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Rev. Mat. Iberoam. 21(1), 179–235 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin (1979) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050)Université Paris-Est CréteilCréteil CedexFrance

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