In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier–Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient which may vanish.
@article{AIHPC_2010__27_5_1227_0, author = {Hmidi, Taoufik and Rousset, Fr\'ed\'eric}, title = {Global well-posedness for the {Navier{\textendash}Stokes{\textendash}Boussinesq} system with axisymmetric data}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1227--1246}, publisher = {Elsevier}, volume = {27}, number = {5}, year = {2010}, doi = {10.1016/j.anihpc.2010.06.001}, zbl = {1200.35229}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.001/} }
TY - JOUR AU - Hmidi, Taoufik AU - Rousset, Frédéric TI - Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1227 EP - 1246 VL - 27 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.001/ DO - 10.1016/j.anihpc.2010.06.001 LA - en ID - AIHPC_2010__27_5_1227_0 ER -
%0 Journal Article %A Hmidi, Taoufik %A Rousset, Frédéric %T Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1227-1246 %V 27 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.001/ %R 10.1016/j.anihpc.2010.06.001 %G en %F AIHPC_2010__27_5_1227_0
Hmidi, Taoufik; Rousset, Frédéric. Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 5, pp. 1227-1246. doi : 10.1016/j.anihpc.2010.06.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.06.001/
[1] Résultats de régularité de solutions axisymétriques pour le système de Navier–Stokes, Bull. Sci. Math. 132 no. 7 (2008), 592-624
,[2] On the global well-posedness for Boussinesq system, J. Differential Equations 233 no. 1 (2007), 199-220 | Zbl
, ,[3] On the global regularity of axisymmetric Navier–Stokes–Boussinesq system, arXiv:0908.0894v1 | Zbl
, , ,[4] On the global well-posedness for the axisymmetric Euler equations, Math. Ann. 347 no. 1 (2010), 15-41 | Zbl
, , ,[5] Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57 (2007), 883-917 | EuDML | Numdam | Zbl
, ,[6] Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), 61-66 | Zbl
, , ,[7] Limite non visqueuse pour les fluides incompressibles axisymétrique, Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, vol. XIV, Paris, 1997/1998, Stud. Math. Appl. vol. 31, North-Holland, Amsterdam (2002), 29-55 | MR
, ,[8] Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, arXiv:0801.1088 (2008) | MR
,[9] First order interpolation inequality with weights, Compos. Math. 53 (1984), 259-275 | EuDML | Numdam | MR | Zbl
, , ,[10] Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math. 203 no. 2 (2006), 497-513 | MR | Zbl
,[11] Perfect Incompressible Fluids, Oxford University Press (1998) | MR
,[12] On the global wellposedness of the 3-D incompressible Navier–Stokes equations, Ann. École Norm. Sup. 39 (2006), 679-698 | EuDML | MR | Zbl
, ,[13] Wellposedness and stability results for the Navier–Stokes equations in , Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 2 (2009), 599-624 | EuDML | Numdam | MR | Zbl
, ,[14] J.-Y. Chemin, I. Gallagher, M. Paicu, Global regularity for some classes of large solutions to the Navier–Stokes equations, Ann. of Math., in press. | MR
[15] Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 no. 3 (2007), 73-94 | MR | Zbl
,[16] Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys. 290 no. 1 (2009), 1-14, arXiv:0806.4081 [math.AP] | MR | Zbl
, ,[17] Le théorème de Leary et le théorème de Fujita–Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France 136 (2008), 261-309 | EuDML | Numdam | MR
, ,[18] Global existence results for the anisotropic Boussinesq system in dimension two, arXiv:0809.4984v1 [math.AP] (2008) | MR
, ,[19] On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations 12 no. 4 (2007), 461-480 | MR | Zbl
, ,[20] Incompressible viscous flows in borderline Besov spaces, Arch. Ration. Mech. Anal. 189 no. 2 (2008), 283-300 | MR | Zbl
, ,[21] On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J. 58 no. 4 (2009), 1591-1618 | MR | Zbl
, ,[22] Global well-posedness for Euler–Boussinesq system, arXiv:0903.3747 (2009) | MR | Zbl
, , ,[23] Global well-posedness for Navier–Stokes–Boussinesq system, arXiv:0904.1536v1 (2009) | MR | Zbl
, , ,[24] Unique solvability in large of a three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry, Zap. Nauchn. Sem. LOMI 7 (1968), 155-177 | Zbl
,[25] Recent Developments in the Navier–Stokes Problem, CRC Press (2002) | MR | Zbl
,[26] On axially symmetric flows in , Z. Anal. Anwend. 18 no. 3 (1999), 639-649 | EuDML | MR | Zbl
, , , ,[27] Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta Math. 63 (1934), 193-248 | MR
,[28] Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci. 70 no. 10 (1994), 299-304 | MR | Zbl
, ,[29] Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Mekh. 32 no. 1 (1968), 59-69 | MR | Zbl
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