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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.bulsci.2007.10.001.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétrique,, in, (2002), 29.   Google Scholar

[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.   Google Scholar

[7]

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

[8]

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

[9]

J.-Y. Chemin, "Perfect Incompressible Fluids,", Oxford University Press 1998., (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

[12]

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

[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.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

[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.   Google Scholar

[15]

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

[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.   Google Scholar

[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.  doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.   Google Scholar

[22]

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

[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.   Google Scholar

[24]

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

[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.  doi: 10.3792/pjaa.70.299.  Google Scholar

[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.   Google Scholar

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.  doi: 10.1016/j.bulsci.2007.10.001.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétrique,, in, (2002), 29.   Google Scholar

[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.   Google Scholar

[7]

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

[8]

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

[9]

J.-Y. Chemin, "Perfect Incompressible Fluids,", Oxford University Press 1998., (1998).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.anihpc.2007.05.008.  Google Scholar

[12]

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

[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.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

[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.   Google Scholar

[15]

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

[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.   Google Scholar

[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.  doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.   Google Scholar

[22]

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

[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.   Google Scholar

[24]

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

[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.  doi: 10.3792/pjaa.70.299.  Google Scholar

[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.   Google Scholar

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