Advertisement

Quasigeostrophic equation with random initial data in negative-order Sobolev space

  • Lihuai Du
  • Huaqiao WangEmail author
Article
  • 4 Downloads

Abstract

This paper is concerned with the Cauchy problem of the Quasigeostrophic equation on the torus \({\mathbb {T}}^2\). We prove the almost sure existence and uniqueness of the global solution with the random initial data in \({\mathcal {H}}^s({\mathbb {T}}^2), s\in [-1,0)\).

Keywords

Quasigeostrophic equation Random initial data Negative-order Sobolev space 

Mathematics Subject Classification

35Q35 76D03 

Notes

Acknowledgements

The authors express their sincere gratitude to the reviewers for their comments and suggestions. H. Wang’s research was supported by National Postdoctoral Program for Innovative Talents (No. BX201600020) and Project No. 2019CDXYST0015 supported by the Fundamental Research Funds for the Central Universities. L. Du’s research was supported by China Postdoctoral Science Foundation (No. 2019M650580).

References

  1. 1.
    Babin, A., Mahalov, A., Nicolaenko, B.: Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids. Asymptot. Anal. 15(2), 103–150 (1997)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bennett, A.F., Kloeden, P.E.: The dissipative quasigeostrophic equations. Mathematika, 28(2):265–285 (1982), 1981MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bourgeois, A.J., Beale, J.T.: Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM J. Math. Anal. 25(4), 1023–1068 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent. Math. 173(3), 449–475 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations. II. A global existence result. Invent. Math. 173(3), 477–496 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burq, N., Tzvetkov, N.: Probabilistic well-posedness for the cubic wave equation. J. Eur. Math. Soc. 16(1), 1–30 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cushman-Roisin, B., Beckers, J.M.: Introduction to Geophysical Fluid Dynamics, vol. 101 of International Geophysics Series, second edition. Elsevier/Academic PressGoogle Scholar
  8. 8.
    Deng, C., Cui, S.B.: Random-data Cauchy problem for the Navier–Stokes equations on \(\mathbb{T}^{3}\). J. Differ. Equ. 251(4–5), 902–917 (2011)CrossRefGoogle Scholar
  9. 9.
    Du, L.H., Zhang, T.: Almost sure existence of global weak solutions for incompressible MHD equations in negative-order Sobolev space. J. Differ. Equ. 263(2), 1611–1642 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Embid, P.F., Majda, A.J.: Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial Differ. Equ. 21(3–4), 619–658 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Holm, D.D.: Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations. Phys. Fluids 29(1), 7–8 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  13. 13.
    Nahmod, A.R., Pavlović, N., Staffilani, G.: Almost sure existence of global weak solutions for supercritical Navier–Stokes equations. SIAM J. Math. Anal. 45(6), 3431–3452 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn. Springer, Berlin (1987)CrossRefGoogle Scholar
  15. 15.
    Schochet, S.: Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation. J. Differ. Equ. 68(3), 400–428 (1987)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhang, T., Fang, D.Y.: Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations. Proc. Am. Math. Soc. 139(8), 2827–2837 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhang, T., Fang, D.Y.: Random data Cauchy theory for the generalized incompressible Navier–Stokes equations. J. Math. Fluid Mech. 14(2), 311–324 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingChina

Personalised recommendations