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Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

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Abstract

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in \({\mathbb{R}^2}\). Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ 0 liying in the space \({\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}\) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation \({(SQG)_{R,\epsilon}}\) which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit (\({R \rightarrow \infty}\), \({\epsilon \rightarrow 0}\)) in \({(SQG)_{R,\epsilon}}\) and that the limit solution has the desired regularity.

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References

  1. Abidi H., Hmidi T.: On the global well-posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal. 40(1), 167–185 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basson A.: Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations. Commun. Math. Phys. 266, 17–35 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Berlin-Heidelberg-New Yark: Springer Verlag, 2011

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Cannone, M.: Ondelettes, paraproduits et Navier-Stokes. Paris: Diderot éditeur, 1995

  6. Chae D., Lee J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003)

    MathSciNet  ADS  MATH  Google Scholar 

  7. Constantin P., Córdoba D., Wu J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J 50, 97–107 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Constantin P., Wu J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937948 (1999)

    Article  MathSciNet  Google Scholar 

  10. Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004)

    Article  ADS  MATH  Google Scholar 

  11. Córdoba D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Anna. Math. 148(3), 1135–1152 (1998)

    Article  MATH  Google Scholar 

  12. Ju N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251(2), 365–376 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Lazar, O.: Global and Local existence for the dissipative critical SQG equation with small oscillations. Preprint

  16. Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem. London: Chapman & Hall/CRC, 2002

  17. Lemarié-Rieusset, P.G.: The Navier-Stokes equations in the critical Morrey-Campanato space Revista Mat. Ibero-amer. 23, 897–930 (2007)

    Google Scholar 

  18. Resnick S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. Thesis, University of Chicago, 1995

  19. Vishik M.I., Fursikov A.V.: Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes. Ann. Scuola Norm. Sup. Pisa, série IV, 531–576 (1977)

    Google Scholar 

  20. Vishik M.I., Fursikov A.V.: Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations. Siberian Math. J. 19, 710–729 (1978)

    Article  MathSciNet  Google Scholar 

  21. Wu J: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal 36, 1014–1030 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France

    Omar Lazar

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  1. Omar Lazar
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Correspondence to Omar Lazar.

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Communicated by P. Constantin

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Lazar, O. Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation. Commun. Math. Phys. 322, 73–93 (2013). https://doi.org/10.1007/s00220-013-1693-2

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  • DOI: https://doi.org/10.1007/s00220-013-1693-2

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