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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 |
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 [
References:
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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. Abidi, T. 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. Beale, T. 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.
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[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. Giga, T. 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. Hmidi, S. 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. Hmidi, S. 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. Larios, E. 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. Liu, M. 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. Abidi, T. 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. Beale, T. 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. Giga, T. 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. Hmidi, S. 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. Hmidi, S. 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. Larios, E. 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. Liu, M. 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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