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

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.

Citation: Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete and 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.

[2]

H. AbidiT. 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. BealeT. 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. BernardiB. 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. HmidiS. 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. HmidiS. 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. AbidiT. 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. BealeT. 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. BernardiB. 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. HmidiS. 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. HmidiS. 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.

[1]

Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1

[2]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397

[3]

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

[4]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[5]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

[6]

Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287

[7]

Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273

[8]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653

[9]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[10]

Christos Sourdis. Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1039-1059. doi: 10.3934/dcds.2017043

[11]

Arnab Roy, Takéo Takahashi. Local null controllability of a rigid body moving into a Boussinesq flow. Mathematical Control and Related Fields, 2019, 9 (4) : 793-836. doi: 10.3934/mcrf.2019050

[12]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055

[13]

Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017

[14]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

[15]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018

[16]

Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025

[17]

Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control and Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121

[18]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509

[19]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153

[20]

Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control and Related Fields, 2022, 12 (2) : 447-473. doi: 10.3934/mcrf.2021030

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (318)
  • HTML views (176)
  • Cited by (1)

Other articles
by authors

[Back to Top]