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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
Article

Abstract

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.

Keywords

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.

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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

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