November  2020, 40(11): 6473-6506. doi: 10.3934/dcds.2020287

On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces

1. 

LEDPA, Université de Batna 2, Faculté des Mathématiques et d'Informatique, Département de Mathématiques, 05000 Batna, Algérie

2. 

Université Côte d'Azur, CNRS, LJAD, Nice, France

* Corresponding author appreciates Taoufik Hmidi from Rennes 1 University for the fruitful discussions on the subject with the occasion of RAMA 11 organized by Sidi bel Abbess University, Algeria, November 2019. A particular greeting dedicates to the referee for the enormous efforts, careful reading and his appreciation of the work

Received  March 2020 Revised  April 2020 Published  July 2020

Fund Project: This work has been done while the second author is a PhD student at the University of Côte d’Azur-Nice-France, under the supervision of F. Planchon and P. Dreyfuss, in particular the second author would like to thank his supervisors, and the LJAD direction

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the initial data $ (v_0, \rho_0) $ is axisymmetric and $ (\omega_0, \rho_0) $ belongs to the critical space $ L^1(\Omega)\times L^1( \mathbb{R}^3) $, with $ \omega_0 $ is the initial vorticity associated to $ v_0 $ and $ \Omega = \{(r, z)\in \mathbb{R}^2:r>0\} $, then the viscous Boussinesq system has a unique global solution.

Citation: 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
References:
[1]

H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. 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. Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.

[3]

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

[4]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$D-Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[5]

M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal., 128 (1994), 329-358.  doi: 10.1007/BF00387712.

[6]

H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal., 128 (1994), 359-360.  doi: 10.1007/BF00387713.

[7]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, in Approximation Methods for Navier-Stokes Problems, Lecture Notes in Math., 771, Springer, Berlin, 1980,129–144. doi: 10.1007/BFb0086903.

[8]

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

[9]

J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations, 21 (1996), 1771-1779.  doi: 10.1080/03605309608821245.

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998. 
[11]

G.-H. Cottet, Equations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105-108. 

[12]

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.

[13]

R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.

[14]

P. Dreyfuss and H. Houamed, Uniqueness result for Navier-Stokes-Boussinesq equations with horizontal dissipation, preprint, arXiv: 1904.00437.

[15]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.

[16]

H. Feng and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal., 215 (2015), 89-123.  doi: 10.1007/s00205-014-0775-4.

[17]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[18]

I. Gallagher and T. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., 332 (2005), 287-327.  doi: 10.1007/s00208-004-0627-x.

[19]

T. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 1091-1131.  doi: 10.3934/dcdss.2012.5.1091.

[20]

T. Gallay and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Math., 7 (2015), 67-92.  doi: 10.5802/cml.25.

[21]

T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.  doi: 10.1007/s00220-004-1254-9.

[22]

P. Germain, Équations de Navier-Stokes dans $ \mathbb{R}^2$: Existence et comportement asymptotique de solutions d'énergie infinie, Bull. Sci. Math., 130 (2006), 123-151.  doi: 10.1016/j.bulsci.2005.06.004.

[23]

Y. GigaT. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.

[24]

B. L. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica, 5 (1989), 208-218.  doi: 10.1007/BF02006004.

[25]

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. 

[26]

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.

[27]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for an Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

[28]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Navier-Stokes-Boussinesq system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[29]

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.

[30]

T. Hmidi and M. Zerguine, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239 (2010), 1387-1401.  doi: 10.1016/j.physd.2009.12.009.

[31]

T. Hmidi and M. Zerguine, Inviscid limit for axisymmetric Navier-Stokes system, Differential Integral Equations, 22 (2009), 1223-1246. 

[32]

T. Hmidi and M. Zerguine, Vortex patch for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  doi: 10.4310/CMS.2014.v12.n8.a8.

[33]

T. Y. 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.

[34]

H. Houamed and M. Zerguine, On the global solvability of the axisymmetric Boussinesq system with critical regularity, Nonlinear Anal., 200 (2020), 26pp. doi: 10.1016/j.na.2020.112003.

[35]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155-177. 

[36]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the $2$D-Boussinesq system with anisotropic viscosity without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.

[37]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[38]

X. LiuM. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.

[39]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.

[40]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. (9), 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.

[41]

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

[42]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.

[43]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781139095143.

[44]

V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$veràk, Selected Topics in Fluid Mechanics. Course notes, (2011).

[45]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-69.  doi: 10.1016/0021-8928(68)90147-0.

[46]

M. Zerguine, The regular vortex patch problem for stratified Euler equations with critical fractional dissipation, J. Evol. Equ., 15 (2015), 667-698.  doi: 10.1007/s00028-015-0277-3.

show all references

References:
[1]

H. Abidi, Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. 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. Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.

[3]

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

[4]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$D-Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[5]

M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal., 128 (1994), 329-358.  doi: 10.1007/BF00387712.

[6]

H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: "Global solutions of two-dimensional Navier-Stokes and Euler equations", Arch. Rational Mech. Anal., 128 (1994), 359-360.  doi: 10.1007/BF00387713.

[7]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, in Approximation Methods for Navier-Stokes Problems, Lecture Notes in Math., 771, Springer, Berlin, 1980,129–144. doi: 10.1007/BFb0086903.

[8]

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

[9]

J.-Y. Chemin, A remark on the inviscid limit for two-dimensional incompressible fluids, Comm. Partial Differential Equations, 21 (1996), 1771-1779.  doi: 10.1080/03605309608821245.

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 14, The Clarendon Press, Oxford University Press, New York, 1998. 
[11]

G.-H. Cottet, Equations de Navier-Stokes dans le plan avec tourbillon initial mesure, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 105-108. 

[12]

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.

[13]

R. Danchin and M. Paicu, Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309.  doi: 10.24033/bsmf.2557.

[14]

P. Dreyfuss and H. Houamed, Uniqueness result for Navier-Stokes-Boussinesq equations with horizontal dissipation, preprint, arXiv: 1904.00437.

[15]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.

[16]

H. Feng and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, On the Cauchy problem for axi-symmetric vortex rings, Arch. Rational Mech. Anal., 215 (2015), 89-123.  doi: 10.1007/s00205-014-0775-4.

[17]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.

[18]

I. Gallagher and T. Gallay, Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity, Math. Ann., 332 (2005), 287-327.  doi: 10.1007/s00208-004-0627-x.

[19]

T. Gallay, Stability and interaction of vortices in two-dimensional viscous flows, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 1091-1131.  doi: 10.3934/dcdss.2012.5.1091.

[20]

T. Gallay and V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$verák, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, Confluentes Math., 7 (2015), 67-92.  doi: 10.5802/cml.25.

[21]

T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys., 255 (2005), 97-129.  doi: 10.1007/s00220-004-1254-9.

[22]

P. Germain, Équations de Navier-Stokes dans $ \mathbb{R}^2$: Existence et comportement asymptotique de solutions d'énergie infinie, Bull. Sci. Math., 130 (2006), 123-151.  doi: 10.1016/j.bulsci.2005.06.004.

[23]

Y. GigaT. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250.  doi: 10.1007/BF00281355.

[24]

B. L. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica, 5 (1989), 208-218.  doi: 10.1007/BF02006004.

[25]

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. 

[26]

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.

[27]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for an Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

[28]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Navier-Stokes-Boussinesq system with critical dissipation, J. Differential Equations, 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[29]

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.

[30]

T. Hmidi and M. Zerguine, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239 (2010), 1387-1401.  doi: 10.1016/j.physd.2009.12.009.

[31]

T. Hmidi and M. Zerguine, Inviscid limit for axisymmetric Navier-Stokes system, Differential Integral Equations, 22 (2009), 1223-1246. 

[32]

T. Hmidi and M. Zerguine, Vortex patch for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  doi: 10.4310/CMS.2014.v12.n8.a8.

[33]

T. Y. 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.

[34]

H. Houamed and M. Zerguine, On the global solvability of the axisymmetric Boussinesq system with critical regularity, Nonlinear Anal., 200 (2020), 26pp. doi: 10.1016/j.na.2020.112003.

[35]

O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 155-177. 

[36]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the $2$D-Boussinesq system with anisotropic viscosity without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.

[37]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.

[38]

X. LiuM. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.

[39]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.

[40]

C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. (9), 101 (2014), 842-872.  doi: 10.1016/j.matpur.2013.10.007.

[41]

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

[42]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.

[43]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781139095143.

[44]

V. ${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over S} }}$veràk, Selected Topics in Fluid Mechanics. Course notes, (2011).

[45]

M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), 52-69.  doi: 10.1016/0021-8928(68)90147-0.

[46]

M. Zerguine, The regular vortex patch problem for stratified Euler equations with critical fractional dissipation, J. Evol. Equ., 15 (2015), 667-698.  doi: 10.1007/s00028-015-0277-3.

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