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Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data. (English) Zbl 1186.35157

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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[1] Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics 34, Cambridge, Cambridge Univ. Press, 1995 · Zbl 0818.47059
[2] Aubin, J.-P.: Un théorème de compacité. Comptes Rendus de l’Académie des Sciences, Paris 256, 5042–5044 (1963) · Zbl 0195.13002
[3] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, to appear · Zbl 1227.35004
[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)
[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 · Zbl 0429.35059
[6] Chae D.: Global regularity for the 2-D Boussinesq equations with partial viscous terms. Adv. Math. 203(2), 497–513 (2006) · Zbl 1100.35084
[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)
[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) · Zbl 1162.35063
[10] E W., Shu C.-W.: Small-scale structures in Boussinesq convection. Phy. Fluids 6(1), 49–58 (1994) · Zbl 0822.76087
[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) · Zbl 0681.76048
[13] Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity, to appear in Indiana University Mathematical Journal · Zbl 1178.35303
[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 · Zbl 1100.76500
[15] Pedlosky J.: Geophysical Fluid Dynamics. Springer Verlag, New-York (1987) · Zbl 0713.76005
[16] Vishik M.: Hydrodynamics in Besov spaces. Arch. Rat. Mech. Anal. 145(3), 197–214 (1998) · Zbl 0926.35123
[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)
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