July  2011, 29(3): 737-756. doi: 10.3934/dcds.2011.29.737

On the global regularity of axisymmetric Navier-Stokes-Boussinesq system

1. 

Faculté des Sciences de Tunis, Tunisia

2. 

IRMAR, Université de Rennes 1, 35042 Rennes, France

3. 

Laboratoire Paul Painlevé, 59655 Villeneuve d'Ascq, France

Received  December 2009 Revised  May 2010 Published  November 2010

In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
Citation: Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737
References:
[1]

H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sc. Math., 132 (2008), 592-624. doi: 10.1016/j.bulsci.2007.10.001.

[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq System, J. Diff. Equa., 233 (2007), 199-220. doi: 10.1016/j.jde.2006.10.008.

[3]

H. Abidi, T. Hmidi and K. Sahbi, On the global well-posedness for the axisymmetric Euler equations, Mathematische Annalen, 347 (2010), 15-41. doi: 10.1007/s00208-009-0425-6.

[4]

H. Abidi and M. Paicu, Existence globale pour un fluide inhomogène, Annales Inst. Fourier, 57 (2007), 883-917.

[5]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétrique, in "Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar. Vol. XIV" (Paris, 1997/1998), Stud. Math. Appl, 31 North. Holland, Amsterdam, (2002), 29-55.

[6]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. de l'Ecole Norm. Sup., 14 (1981), 209-246.

[7]

Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci., 19 (2009), 547-570. doi: 10.1007/s00332-009-9044-3.

[8]

D. Chae, Global regularity for the $2$-D Boussinesq equations with partial viscous terms, Advances in Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.

[9]

J.-Y. Chemin, "Perfect Incompressible Fluids," Oxford University Press 1998.

[10]

J.-Y. Chemin and I. Gallagher, On the global wellposedness of the $3$-D incompressible Navier-Stokes equations with large initial data, Ann. de l'Ecole Norm. Sup., 39 (2006), 679-698.

[11]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $R^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008.

[12]

R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys, 62 (2007), 73-94.

[13]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.

[14]

R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bulletin de la S. M. F., 136 (2008), 261-309.

[15]

R. Danchin and M. Paicu, Global existence results for the anistropic Boussinesq system in dimension two,, preprint, (). 

[16]

T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.

[17]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.

[18]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system,, preprint, (). 

[19]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Navier-Stokes-Boussinesq system,, preprint, (). 

[20]

R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.

[21]

O. A. Ladyzhenskaya, Unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zapisky Nauchnych Sem. LOMI, 7 (1968), 155-177.

[22]

P.-G. Lemarié, "Recent Developments in the Navier-Stokes Problem," CRC Press, 2002. doi: 10.1201/9781420035674.

[23]

S. Leonardi, J. Málek, J. Necăs and M. Pokorný, On axially symmetric flows in $\RR^3$, Zeitschrift für Analysis und ihre Anwendungen [Journal for Analysis and its Applications Volume], 18 (1999), 639-649.

[24]

J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta mathematica, 63 (1934), 193-248. doi: 10.1007/BF02547354.

[25]

T. Shirota and T. Yanagisawa, Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 299-304. doi: 10.3792/pjaa.70.299.

[26]

M. R. Ukhovskii, V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32 (1968), 59-69.

show all references

References:
[1]

H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sc. Math., 132 (2008), 592-624. doi: 10.1016/j.bulsci.2007.10.001.

[2]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq System, J. Diff. Equa., 233 (2007), 199-220. doi: 10.1016/j.jde.2006.10.008.

[3]

H. Abidi, T. Hmidi and K. Sahbi, On the global well-posedness for the axisymmetric Euler equations, Mathematische Annalen, 347 (2010), 15-41. doi: 10.1007/s00208-009-0425-6.

[4]

H. Abidi and M. Paicu, Existence globale pour un fluide inhomogène, Annales Inst. Fourier, 57 (2007), 883-917.

[5]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétrique, in "Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar. Vol. XIV" (Paris, 1997/1998), Stud. Math. Appl, 31 North. Holland, Amsterdam, (2002), 29-55.

[6]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. de l'Ecole Norm. Sup., 14 (1981), 209-246.

[7]

Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci., 19 (2009), 547-570. doi: 10.1007/s00332-009-9044-3.

[8]

D. Chae, Global regularity for the $2$-D Boussinesq equations with partial viscous terms, Advances in Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.

[9]

J.-Y. Chemin, "Perfect Incompressible Fluids," Oxford University Press 1998.

[10]

J.-Y. Chemin and I. Gallagher, On the global wellposedness of the $3$-D incompressible Navier-Stokes equations with large initial data, Ann. de l'Ecole Norm. Sup., 39 (2006), 679-698.

[11]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $R^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624. doi: 10.1016/j.anihpc.2007.05.008.

[12]

R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys, 62 (2007), 73-94.

[13]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14. doi: 10.1007/s00220-009-0821-5.

[14]

R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bulletin de la S. M. F., 136 (2008), 261-309.

[15]

R. Danchin and M. Paicu, Global existence results for the anistropic Boussinesq system in dimension two,, preprint, (). 

[16]

T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.

[17]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618. doi: 10.1512/iumj.2009.58.3590.

[18]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system,, preprint, (). 

[19]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Navier-Stokes-Boussinesq system,, preprint, (). 

[20]

R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.

[21]

O. A. Ladyzhenskaya, Unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zapisky Nauchnych Sem. LOMI, 7 (1968), 155-177.

[22]

P.-G. Lemarié, "Recent Developments in the Navier-Stokes Problem," CRC Press, 2002. doi: 10.1201/9781420035674.

[23]

S. Leonardi, J. Málek, J. Necăs and M. Pokorný, On axially symmetric flows in $\RR^3$, Zeitschrift für Analysis und ihre Anwendungen [Journal for Analysis and its Applications Volume], 18 (1999), 639-649.

[24]

J. Leray, Sur le mouvement d'un liquide visqueux remplissant l'espace, Acta mathematica, 63 (1934), 193-248. doi: 10.1007/BF02547354.

[25]

T. Shirota and T. Yanagisawa, Note on global existence for axially symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 299-304. doi: 10.3792/pjaa.70.299.

[26]

M. R. Ukhovskii, V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32 (1968), 59-69.

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