no. 5
Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1227-1246.

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 κ⩾0 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://www.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://www.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://www.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, Tome 27 (2010) no. 5, pp. 1227-1246. doi : 10.1016/j.anihpc.2010.06.001. http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.001/

[1] H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier–Stokes, Bull. Sci. Math. 132 no. 7 (2008), 592-624

[2] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 no. 1 (2007), 199-220 | Zbl

[3] H. Abidi, T. Hmidi, K. Sahbi, On the global regularity of axisymmetric Navier–Stokes–Boussinesq system, arXiv:0908.0894v1 | Zbl

[4] H. Abidi, T. Hmidi, K. Sahbi, On the global well-posedness for the axisymmetric Euler equations, Math. Ann. 347 no. 1 (2010), 15-41 | Zbl

[5] H. Abidi, M. Paicu, Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57 (2007), 883-917 | EuDML | Numdam | Zbl

[6] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), 61-66 | Zbl

[7] J. Ben Ameur, R. Danchin, 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] Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, arXiv:0801.1088 (2008) | MR

[9] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984), 259-275 | EuDML | Numdam | MR | Zbl

[10] D. Chae, Global regularity for the 2-D Boussinesq equations with partial viscous terms, Adv. Math. 203 no. 2 (2006), 497-513 | MR | Zbl

[11] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press (1998) | MR

[12] J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D incompressible Navier–Stokes equations, Ann. École Norm. Sup. 39 (2006), 679-698 | EuDML | MR | Zbl

[13] J.-Y. Chemin, I. Gallagher, Wellposedness and stability results for the Navier–Stokes equations in R 3 , 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] R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 no. 3 (2007), 73-94 | MR | Zbl

[16] R. Danchin, M. Paicu, 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] R. Danchin, M. Paicu, 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] R. Danchin, M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, arXiv:0809.4984v1 [math.AP] (2008) | MR

[19] T. Hmidi, S. Keraani, 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] T. Hmidi, S. Keraani, Incompressible viscous flows in borderline Besov spaces, Arch. Ration. Mech. Anal. 189 no. 2 (2008), 283-300 | MR | Zbl

[21] T. Hmidi, S. Keraani, 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] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for Euler–Boussinesq system, arXiv:0903.3747 (2009) | MR | Zbl

[23] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for Navier–Stokes–Boussinesq system, arXiv:0904.1536v1 (2009) | MR | Zbl

[24] O.A. Ladyzhenskaya, 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] P.-G. Lemarié, Recent Developments in the Navier–Stokes Problem, CRC Press (2002) | MR | Zbl

[26] S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in ℝ 3 , Z. Anal. Anwend. 18 no. 3 (1999), 639-649 | EuDML | MR | Zbl

[27] J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta Math. 63 (1934), 193-248 | MR

[28] T. Shirota, T. Yanagisawa, 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] M.R. Ukhovskii, V.I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Mekh. 32 no. 1 (1968), 59-69 | MR | Zbl

Cité par Sources :