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.


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
Full Text: DOI arXiv


[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 · doi:10.1016/j.aim.2005.05.001
[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 · doi:10.1063/1.868044
[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 · doi:10.1007/BF02006004
[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 · doi:10.1007/s002050050128
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.