Communications in Mathematical Physics

, Volume 319, Issue 1, pp 195–229 | Cite as

Global Well-Posedness of an Inviscid Three-Dimensional Pseudo-Hasegawa-Mima Model

  • Chongsheng Cao
  • Aseel Farhat
  • Edriss S. TitiEmail author


The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-Bénard convection model. The mathematical analysis of the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. This is in addition to proving and implementing a new technical logarithmic inequality, generalizing the Brezis-Gallouet and the Brezis-Wainger inequalities. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations.


Vorticity Weak Solution Euler Equation Global Existence Global Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akerstedt H.O., Nycander J., Pavlenko V.P.: Three-dimensional stability of drift vortices. Phys. Plasmas 3(1), 160–167 (1996)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Brézis H., Gallouet T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Brézis H., Wainger S.: A Note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Par. Diff. Eqs. 5(7), 773–789 (1980)zbMATHCrossRefGoogle Scholar
  4. 4.
    Cao C., Wu J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226(2), 1803–1822 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 1988Google Scholar
  6. 6.
    Danchin R., Paicu M.: Global existence results for the anisotropic Boussinesq system in dimension two. Math. Models Meths. Appl. Sci. 22(3), 421–457 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, Providence, RI: Amer. Math. Soc., 1998Google Scholar
  8. 8.
    Farhat, A., Hauk, S., Titi, E.S.: Analytical study of the Stommel-Charney model of the gulf stream and its relation to the two-dimensional Hasegawa-Mima equation, PreprintGoogle Scholar
  9. 9.
    Foias C., Manley O., Temam R.: Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal. 11(8), 939–967 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gao H., Zhu A.: The global strong solutions of Hasegawa-Mima- Charney-Obukhov equation. J. Math. Phys. 46(8), 083517 (2005)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Grauer R.: An energy estimate for a perturbed Hasegawa-Mima equation. Nonlinearity 11(3), 659–666 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Guo B., Han Y.: Existence and uniqueness of global solution of the Hasegawa-Mima equation. J. Math. Phys. 45(4), 1639–1647 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Guo, B., Huang, D.: Existence and stability of steady waves for the Hasegawa-Mima Equation. Bound. Value Probl. 2009, Art. ID 509801 (2009)Google Scholar
  14. 14.
    Hasegawa A., Mima K.: Stationary spectrum of strong turbulence in magnetized nonuniform plasma. Phys. Rev. Lett. 39(4), 206–208 (1977)ADSCrossRefGoogle Scholar
  15. 15.
    Hasegawa A., Mima K.: Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids 21(1), 87–92 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Hasegawa A., Mima K.: Nonlinear instability of electromagnetic drift waves. Phys. Fluids 21(1), 81–86 (1978)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Hasegawa A., Wakatani M.: Plasma edge turbulence. Phys. Rev. Lett. 50(9), 682–686 (1983)ADSCrossRefGoogle Scholar
  18. 18.
    Horton W., Meiss J.D.: Solitary drift waves in the presence of magnetic shear. Phys. Fluids 26(4), 990–997 (1983)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Hou T.Y., Li C.: Global well-posedness of the viscous Boussinesq equations. Dis. Cont. Dyn. Syst. 12(1), 1–12 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    John, F.: Partial Differential Equations. Applied Mathematical Sciences, Vol. 19, New York: Springer-Verlag, 1986Google Scholar
  21. 21.
    Julien K., Knobloch E., Milliff R., Werne J.: Generalized quasigeostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Kupferman R., Mangoubi C., Titi E.S.: A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime. Commun. Math. Sci. 6(1), 235–256 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, Vol. 27, Cambridge: Cambridge University Press, 2002Google Scholar
  24. 24.
    Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, Vol. 96, New York: Springer-Verlag, 1994Google Scholar
  25. 25.
    Nicolaenko B., Scheurer B., Temam R.: Some global dynamical properties of the Kuramoto–Sivashinsky equation: Nonlinear stability and attractors. Physica D 16, 155–183 (1985)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Paumond L.: Some remarks on a Hasegawa-Mima-Charney-Obukhov equation. Phys. D 195(3-4), 379–390 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Pedlosky J.: The equations for geostrophic motion in the ocean. J. Phys. Oceanogr. 14, 448–455 (1984)ADSCrossRefGoogle Scholar
  28. 28.
    Pedlosky J.: Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987Google Scholar
  29. 29.
    Sprague M., Julien K., Knobloch E., Werne J.: Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141–174 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Sueyoshi M., Iwayama T.: Hamiltonian structure for the Charney-Hasegawa-Mima equation in the asymptotic model regime. Fluid Dynam. Res. 39(4), 346–352 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Tassi E., Chandre C., Morrison P.J.: Hamiltonian derivation of the Charney-Hasegawa-Mima equation. Phys. Plasmas 16, 082301 (2009)ADSCrossRefGoogle Scholar
  32. 32.
    Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edition, Applied Mathematical Sciences, Vol. 68, New York: Springer-Verlag, 1997Google Scholar
  33. 33.
    Temam R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS/Chelsea Publishing, Providence, RI: Amer. Math. Soc., 2001, reprint of the 1984 editionGoogle Scholar
  34. 34.
    Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 6, 1407–1456 (1963) (English)Google Scholar
  35. 35.
    Zhang, P.: Global smooth solutions to the 2-D nonhomogeneous Navier-Stokes equations. Int. Math. Res. Not. IMRN 2008, Art. ID rnn 098 (2008)Google Scholar
  36. 36.
    Zhang R.: The global attractors for the dissipative generalized Hasegawa-Mima equation. Acta Math. Appl. Sin. Engl. Ser. 24(1), 19–28 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zhang R., Guo B.: Global attractor for the Hasegawa-Mima equation. Appl. Math. Mech. 27(5), 505–511 (2006)MathSciNetADSGoogle Scholar
  38. 38.
    Zhang R., Guo B.: Dynamical behavior for the three dimensional generalized Hasegawa-Mima equations. J. Math. Phys. 48(1), 012703 (2007)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Chongsheng Cao
    • 1
  • Aseel Farhat
    • 2
  • Edriss S. Titi
    • 2
    • 3
    • 4
    • 5
    Email author
  1. 1.Department of MathematicsFlorida International UniversityMiamiUSA
  2. 2.Department of MathematicsUniversity of CaliforniaIrvineUSA
  3. 3.Department of Mechanical and Aero-space EngineeringUniversity of CaliforniaIrvineUSA
  4. 4.Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  5. 5.Fellow of the Center of Smart Interfaces (CSI)Technische Universität DarmstadtDarmstadtGermany

Personalised recommendations