Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The Inviscid Three Dimensional Quasi-Geostrophic System on Bounded Domains


We present a formal derivation of the inviscid three dimensional quasi-geostrophic system (QG) from primitive equations on a bounded, cylindrical domain. A key point in the derivation is the treatment of the lateral boundary and the resulting boundary conditions it imposes on solutions. To our knowledge, these boundary conditions are new and differentiate our model from closely related models which have been the object of recent study. These boundary conditions are natural for a variational problem in a particular Hilbert space. We construct solutions and prove an elliptic regularity theorem corresponding to the variational problem, allowing us to show the existence of global weak solutions to QG.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aubin, J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris256, 5042–5044, 1963

  2. 2.

    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin. 1976. Grundlehren der Mathematischen Wissenschaften, No. 223

  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

  4. 4.

    Buckmaster, T., Shkoller, S., Vicol, V.: Nonuniqueness of weak solutions to the SQG equation. 2016. ArXiv e-prints

  5. 5.

    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171(3), 1903–1930, 2010

  6. 6.

    Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-d quasigeostrophic thermal active scalar. Nonlinearity7(6), 1495–1533, 1994

  7. 7.

    Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22(5), 1289–1321, 2012

  8. 8.

    Constantin, P., Ignatova , M.: Critical SQG in bounded domains. Ann. PDE2(2), 8, 2016

  9. 9.

    Constantin, P., Ignatova, M.: Remarks on the fractional laplacian with Dirichlet boundary conditions and applications. Int. Math. Res. Not. 2017(6), 1653–1673, 2017

  10. 10.

    Constantin, P., Ignatova, M., Nguyen, H.Q.: Inviscid limit for SQG in bounded domains. SIAM J. Math. Anal. 50(6), 6196–6207, 2018

  11. 11.

    Constantin, P., Nguyen, H.Q.: Global weak solutions for SQG in bounded domains. Commun. Pure Appl. Math. 2017a

  12. 12.

    Constantin, P., Nguyen, H.Q.: Local and global strong solutions for SQG in bounded domains. Phys. D Nonlinear Phenom. 2017b

  13. 13.

    Desjardins, B., Grenier, E.: Derivation of quasi-geostrophic potential vorticity equations. Adv. Differ. Equ. 3(5), 715–752, 1998

  14. 14.

    Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence 2010

  15. 15.

    Kiselev, A., Nazarov, F.: Variation on a theme of caffarelli and vasseur. J. Math. Sci. 166(1), 31–39, 2010

  16. 16.

    Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2d dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445–453, 2007

  17. 17.

    Marchand, F.: Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces \(L^p\) or \({\dot{H}}^{-\frac{1}{2}}\). Commun. Math. Phys. 277(1), 45–67, 2008

  18. 18.

    Novack, M.D.: On the Weak Solutions to the 3D Inviscid Quasi-Geostrophic System. 2017. ArXiv e-prints

  19. 19.

    Novack, M.: Non-uniqueness of Weak Solutions to the 3D Quasi-Geostrophic Equations. 2018. arXiv e-prints, arXiv:1812.08734

  20. 20.

    Novack, M., Vasseur, A.: Classical Solutions for the 3D Quasi-Geostrophic System on a Bounded Domain. 2019. arXiv e-prints, page arXiv:1904.13050

  21. 21.

    Novack, M.D., Vasseur, A.F.: Global in time classical solutions to the 3d quasi-geostrophic system for large initial data. Commun. Math. Phys. 358(1), 237–267, 2017

  22. 22.

    Puel, M., Vasseur, A.: Global weak solutions to the inviscid 3D quasi-geostrophic equation. Commun. Math. Phys. 339(3), 1063–1082, 2015

  23. 23.

    Resnick, S.: Dynamical Problems in Non-linear Advective Partial Differential Equations. PhD thesis, University of Chicago, 1995

  24. 24.

    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978

  25. 25.

    Triebel, H.: Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Rev. Mat. Complut. 15(2), 475–524, 2002

Download references


The Alexis F. Vasseur was partially funded by the NSF (Grant No. 1209420) during this work.

Author information

Correspondence to Matthew D. Novack.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by P. Constantin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Novack, M.D., Vasseur, A.F. The Inviscid Three Dimensional Quasi-Geostrophic System on Bounded Domains. Arch Rational Mech Anal 235, 973–1010 (2020). https://doi.org/10.1007/s00205-019-01437-x

Download citation