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April  2019, 39(4): 1613-1650. doi: 10.3934/dcds.2019072

On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data

Saoussen Sokrani

Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar 2092 Tunis, Tunisia

Received  June 2017 Revised  June 2018 Published  January 2019

In this paper we prove the global well-posedness for the Three dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with vanishing the horizontal viscosity with a transport-diffusion equation governing the temperature.

Keywords: Boussinesq systems, Littlewood-Paley theory, axisymmetric flow.
Mathematics Subject Classification: 35Q30, 76D03.
Citation: Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072
References:
[1]

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

[2]

H. Abidi, T. Hmidi and S. Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dym. Syst., 29 (2011), 737-756.  doi: 10.3934/dcds.2011.29.737.  Google Scholar

[3]

H. Abidi and P. Marius, On the global well-posedness of 3-D Navier-Stokes equations with vanishing horizontal viscosity, Differential and Integral Equations, 31 (2018), 329-352.   Google Scholar

[4]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[5]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler Equations, Commun. Math. Phys, 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[6]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétriques, in Nonlinear Partial Differential Equations and their Applications: Coll ge de France Seminar Volume XIV (eds. D. Cioranescu and J.-L. Lions), Elsevier, Academic Press, Stud. Math. Appl, 31, North. Holland, Amsterdam, (2002), 29–55. doi: 10.1016/S0168-2024(02)80004-2.  Google Scholar

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C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par élément finis, RAIRO Modél. Math. Anal. Numér, 29 (1995), 871-921.  doi: 10.1051/m2an/1995290708711.  Google Scholar

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J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[9]

J. Boussinesq, Théorie Analytique De La Chaleur, Gauthier-Villars, Paris, 1903. Google Scholar

[10]

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

[11]

M. Cwikel, On $\bigl(L^{p_0}(A_0),L^{p_1}(A_1)\bigr)_{\theta,q}$, Proc. Amer. Math. Soc., 44 (1974), 286-292.  doi: 10.1090/S0002-9939-1974-0358326-0.  Google Scholar

[12]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Physica D., 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.  Google Scholar

[13]

R. Danchin and M. Paicu, Les théorèmes de Leray de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. math. France., 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.  Google Scholar

[14]

R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russ. Math. Surv., 62 (2007), 475-496.  doi: 10.4213/rm6761.  Google Scholar

[15]

E. Feireisl and A. Novotny, The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system, J. Math. Fluid. Mech., 11 (2009), 274-302.  doi: 10.1007/s00021-007-0259-5.  Google Scholar

[16] T. M. Fleet, Differential Analysis, Cambridge University Press, 1980.   Google Scholar
[17]

L. Grafakos, Classical Fourier Analysis, 2nd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-09432-8.  Google Scholar

[18]

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.  Google Scholar

[19]

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.   Google Scholar

[20]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, comm, Part. Diff. Eqs., 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

[21]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Boussinesq-Navier-Stokes system with critical dissipation, J. Diff. Eqs., 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[22]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Functional Analysis, 260 (2011), 745-796.  doi: 10.1016/j.jfa.2010.10.012.  Google Scholar

[23]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire., 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[24]

T. Hou and C. Li, Global well- posedness of the viscous Boussinesq equations, Discrete contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

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

[26]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, 1nd edition, Chapman & Hall/CRC Research Notes in Mathematics, 2002. doi: 10.1201/9781420035674.  Google Scholar

[27]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[28]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl., 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.  Google Scholar

[29]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Commun. Math. Phys., 321 (2013), 33-67.  doi: 10.1007/s00220-013-1721-2.  Google Scholar

[30] A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, 2 edition, Cambridge University Press, New York, 2005.  doi: 10.1017/CBO9780511755422.  Google Scholar
[31]

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.  Google Scholar

[32]

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

[33]

H. Triebel, Interpolation theory, function spaces, differential operators, Bull. Amer. Math. Soc., 2 (1980), 339-345.   Google Scholar

[34]

M. Ukhovskii and V. Yudovitch, Axially symmetric flows of ideal and viscous fluids filling the whole space, Journal of applied mathematics and mechanics., 32 (1968), 52-61.  doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

show all references

References:
[1]

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

[2]

H. Abidi, T. Hmidi and S. Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dym. Syst., 29 (2011), 737-756.  doi: 10.3934/dcds.2011.29.737.  Google Scholar

[3]

H. Abidi and P. Marius, On the global well-posedness of 3-D Navier-Stokes equations with vanishing horizontal viscosity, Differential and Integral Equations, 31 (2018), 329-352.   Google Scholar

[4]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[5]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler Equations, Commun. Math. Phys, 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[6]

J. Ben Ameur and R. Danchin, Limite non visqueuse pour les fluides incompressibles axisymétriques, in Nonlinear Partial Differential Equations and their Applications: Coll ge de France Seminar Volume XIV (eds. D. Cioranescu and J.-L. Lions), Elsevier, Academic Press, Stud. Math. Appl, 31, North. Holland, Amsterdam, (2002), 29–55. doi: 10.1016/S0168-2024(02)80004-2.  Google Scholar

[7]

C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: Le modèle et son approximation par élément finis, RAIRO Modél. Math. Anal. Numér, 29 (1995), 871-921.  doi: 10.1051/m2an/1995290708711.  Google Scholar

[8]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[9]

J. Boussinesq, Théorie Analytique De La Chaleur, Gauthier-Villars, Paris, 1903. Google Scholar

[10]

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

[11]

M. Cwikel, On $\bigl(L^{p_0}(A_0),L^{p_1}(A_1)\bigr)_{\theta,q}$, Proc. Amer. Math. Soc., 44 (1974), 286-292.  doi: 10.1090/S0002-9939-1974-0358326-0.  Google Scholar

[12]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Physica D., 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.  Google Scholar

[13]

R. Danchin and M. Paicu, Les théorèmes de Leray de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. math. France., 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.  Google Scholar

[14]

R. Danchin, Axisymmetric incompressible flows with bounded vorticity, Russ. Math. Surv., 62 (2007), 475-496.  doi: 10.4213/rm6761.  Google Scholar

[15]

E. Feireisl and A. Novotny, The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system, J. Math. Fluid. Mech., 11 (2009), 274-302.  doi: 10.1007/s00021-007-0259-5.  Google Scholar

[16] T. M. Fleet, Differential Analysis, Cambridge University Press, 1980.   Google Scholar
[17]

L. Grafakos, Classical Fourier Analysis, 2nd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-09432-8.  Google Scholar

[18]

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.  Google Scholar

[19]

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.   Google Scholar

[20]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, comm, Part. Diff. Eqs., 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

[21]

T. Hmidi, S. Keraani and F. Rousset, Global well-posedness for Boussinesq-Navier-Stokes system with critical dissipation, J. Diff. Eqs., 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.  Google Scholar

[22]

T. Hmidi and F. Rousset, Global well-posedness for the Euler-Boussinesq system with axisymmetric data, J. Functional Analysis, 260 (2011), 745-796.  doi: 10.1016/j.jfa.2010.10.012.  Google Scholar

[23]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire., 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[24]

T. Hou and C. Li, Global well- posedness of the viscous Boussinesq equations, Discrete contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

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

[26]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, 1nd edition, Chapman & Hall/CRC Research Notes in Mathematics, 2002. doi: 10.1201/9781420035674.  Google Scholar

[27]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[28]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl., 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.  Google Scholar

[29]

C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Commun. Math. Phys., 321 (2013), 33-67.  doi: 10.1007/s00220-013-1721-2.  Google Scholar

[30] A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, 2 edition, Cambridge University Press, New York, 2005.  doi: 10.1017/CBO9780511755422.  Google Scholar
[31]

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.  Google Scholar

[32]

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

[33]

H. Triebel, Interpolation theory, function spaces, differential operators, Bull. Amer. Math. Soc., 2 (1980), 339-345.   Google Scholar

[34]

M. Ukhovskii and V. Yudovitch, Axially symmetric flows of ideal and viscous fluids filling the whole space, Journal of applied mathematics and mechanics., 32 (1968), 52-61.  doi: 10.1016/0021-8928(68)90147-0.  Google Scholar

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