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On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data
Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar 2092 Tunis, Tunisia |
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.
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. Boussinesq, Théorie Analytique De La Chaleur, Gauthier-Villars, Paris, 1903. |
[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. |
[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. |
[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. |
[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. |
[14] |
R. Danchin,
Axisymmetric incompressible flows with bounded vorticity, Russ. Math. Surv., 62 (2007), 475-496.
doi: 10.4213/rm6761. |
[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. |
[16] |
T. M. Fleet, Differential Analysis, Cambridge University Press, 1980.
![]() ![]() |
[17] |
L. Grafakos, Classical Fourier Analysis, 2nd edition, Springer, New York, 2008.
doi: 10.1007/978-0-387-09432-8. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[27] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[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. |
[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. |
[30] |
A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, 2 edition, Cambridge University Press, New York, 2005.
doi: 10.1017/CBO9780511755422.![]() ![]() ![]() |
[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. |
[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. |
[33] |
H. Triebel,
Interpolation theory, function spaces, differential operators, Bull. Amer. Math. Soc., 2 (1980), 339-345.
|
[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. |
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
J. Boussinesq, Théorie Analytique De La Chaleur, Gauthier-Villars, Paris, 1903. |
[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. |
[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. |
[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. |
[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. |
[14] |
R. Danchin,
Axisymmetric incompressible flows with bounded vorticity, Russ. Math. Surv., 62 (2007), 475-496.
doi: 10.4213/rm6761. |
[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. |
[16] |
T. M. Fleet, Differential Analysis, Cambridge University Press, 1980.
![]() ![]() |
[17] |
L. Grafakos, Classical Fourier Analysis, 2nd edition, Springer, New York, 2008.
doi: 10.1007/978-0-387-09432-8. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[27] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant lespace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[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. |
[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. |
[30] |
A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics, 2 edition, Cambridge University Press, New York, 2005.
doi: 10.1017/CBO9780511755422.![]() ![]() ![]() |
[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. |
[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. |
[33] |
H. Triebel,
Interpolation theory, function spaces, differential operators, Bull. Amer. Math. Soc., 2 (1980), 339-345.
|
[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. |
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