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Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

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Abstract

The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. No smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Bénard system are also discussed.

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References

  1. Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics 34, Cambridge, Cambridge Univ. Press, 1995

  2. Aubin, J.-P.: Un théorème de compacité. Comptes Rendus de l’Académie des Sciences, Paris 256, 5042–5044 (1963)

    Google Scholar 

  3. Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, to appear

  4. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Scie. de l’école Normale Sup., 14, 209–246 (1981)

    Google Scholar 

  5. Cannon, J.R., Dibenedetto, E.: The Initial Value Problem for the Boussinesq Equations with Data in L p, Lecture Notes in Math. 771, Berlin-Heidelberg-New York: Springer, 1980, pp. 129–144

  6. Chae D.: Global regularity for the 2-D Boussinesq equations with partial viscous terms. Adv. Math. 203(2), 497–513 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chemin, J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 1995

  8. Chemin J.-Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. d’Anal. Math. 77, 25–50 (1999)

    MathSciNet  Google Scholar 

  9. Danchin R., Paicu M.: Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux. Bull. So. Math. France 136(2), 261–309 (2008)

    MATH  MathSciNet  Google Scholar 

  10. E W., Shu C.-W.: Small-scale structures in Boussinesq convection. Phy. Fluids 6(1), 49–58 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Gérard, P.: Résultats récents sur les fluides parfaits incompressibles bidimensionnels (d’après J.-Y. Chemin et J.-M. Delort). Séminaire Bourbaki, Vol. 1991/92, Astérisque 206, 411–444 (1992)

  12. Guo B.: Spectral method for solving two-dimensional Newton-Boussineq equation. Acta Math. Appl. Sinica 5, 27–50 (1989)

    Article  MATH  Google Scholar 

  13. Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity, to appear in Indiana University Mathematical Journal

  14. Moffatt, H.K.: Some remarks on topological fluid mechanics. In: An Introduction to the Geometry and Topology of Fluid Flows. R. L. Ricca, ed., Dordrecht: Kluwer Academic Publishers, 2001, pp. 3–10

  15. Pedlosky J.: Geophysical Fluid Dynamics. Springer Verlag, New-York (1987)

    MATH  Google Scholar 

  16. Vishik M.: Hydrodynamics in Besov spaces. Arch. Rat. Mech. Anal. 145(3), 197–214 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yudovich V.: Non-stationary flows of an ideal incompressible fluid. Akademija Nauk SSSR. Žurnal Vyčislitel’noĭ Matematiki i Matematičeskoĭ Fiziki 3, 1032–1066 (1963)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, 61 avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Raphaël Danchin

  2. Université Paris 11, Laboratoire de Mathématiques, Bâtiment 425, 91405, Orsay Cedex, France

    Marius Paicu

Authors
  1. Raphaël Danchin
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  2. Marius Paicu
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Correspondence to Raphaël Danchin.

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Communicated by P. Constantin

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Danchin, R., Paicu, M. Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data. Commun. Math. Phys. 290, 1–14 (2009). https://doi.org/10.1007/s00220-009-0821-5

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  • DOI: https://doi.org/10.1007/s00220-009-0821-5

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