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One component regularity criteria for the axially symmetric MHD-Boussinesq system
1. | School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China |
2. | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China |
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $ T_\ast $ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field or the temperature fluctuation is imposed.
References:
[1] |
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. |
[2] |
H. Beirão da Veiga,
A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.
|
[3] |
D. Bian and X. Pu, Global smooth axisymmetic solutions of the Boussinesq equations for magnetohydrodynamics convection, J. Math. Fluid Mech., 22 (2020), Article number: 12, 13pp.
doi: 10.1007/s00021-019-0468-8. |
[4] |
C. Cao and J. Wu,
Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.
doi: 10.1007/s00205-013-0610-3. |
[5] |
D. Chae,
Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.
doi: 10.1016/j.aim.2005.05.001. |
[6] |
D. Chae and J. Lee,
On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671.
doi: 10.1007/s002090100317. |
[7] |
H. Chen, D. Fang and T. Zhang,
Regularity of 3D axisymmetric Navier-Stokes equations, Discrete Contin. Dyn. Syst., 37 (2017), 1923-1939.
doi: 10.3934/dcds.2017081. |
[8] |
Q. Chen, C. Miao and Z. Zhang,
On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.
doi: 10.1007/s00220-008-0545-y. |
[9] |
Q. Chen and Z. Zhang,
Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, J. Math. Anal. Appl., 331 (2007), 1384-1395.
doi: 10.1016/j.jmaa.2006.09.069. |
[10] |
E. B. Fabes, B. F. Jones and N. M. Rivière,
The initial value problem for the Navier-Stokes equations with data in ${L}^p$, Arch. Rational Mech. Anal., 45 (1972), 222-240.
doi: 10.1007/BF00281533. |
[11] |
Y. Giga,
Solutions for semilinear parabolic equations in ${L}^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[12] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[13] |
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. |
[14] |
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. |
[15] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[16] |
H. Kozono and Y. Taniuchi,
Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys., 214 (2000), 191-200.
doi: 10.1007/s002200000267. |
[17] |
A. Larios, E. Lunasin and E. S. Titi,
Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.
doi: 10.1016/j.jde.2013.07.011. |
[18] |
A. Larios and Y. Pei,
On the local well-posedness and a Prodi-Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion, J. Differential Equations, 263 (2017), 1419-1450.
doi: 10.1016/j.jde.2017.03.024. |
[19] |
Z. Lei,
On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j.jde.2015.04.017. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, CRC Press, 2002.
doi: 10.1201/9781420035674.![]() ![]() ![]() |
[21] |
F. Lin, L. Xu and P. Zhang,
Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.
doi: 10.1016/j.jde.2015.06.034. |
[22] |
H. Liu, D. Bian and X. Pu, Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion, Z. Angew. Math. Phys., 70 (2019), Article number: 81, 19pp.
doi: 10.1007/s00033-019-1126-y. |
[23] |
A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, vol. 9 of Courant Lecture Notes in Mathematics, AMS/CIMS, 2003.
doi: 10.1090/cln/009. |
[24] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Appl. Math. Sci., Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
[25] |
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. |
[26] |
G. Mulone and S. Rionero,
Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem, Arch. Ration. Mech. Anal., 166 (2003), 197-218.
doi: 10.1007/s00205-002-0230-9. |
[27] |
X. Pan, Global regularity of solutions for the 3D non-resistive and non-diffusive MHD-Boussinesq system with axisymmetric data, preprint, arXiv: 1911.01550v2. |
[28] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
[29] |
G. Prodi,
Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[30] |
X. Ren, J. Wu, Z. Xiang and Z. Zhang,
Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.
doi: 10.1016/j.jfa.2014.04.020. |
[31] |
J. Serrin,
On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.
doi: 10.1007/BF00253344. |
[32] |
J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, (1963), 69–98. |
[33] |
M. Struwe,
On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.
doi: 10.1002/cpa.3160410404. |
[34] |
S. Takahashi,
On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254.
doi: 10.1007/BF02567922. |
show all references
References:
[1] |
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. |
[2] |
H. Beirão da Veiga,
A new regularity class for the Navier-Stokes equations in $\mathbb{R}^n$, Chinese Ann. Math. Ser. B, 16 (1995), 407-412.
|
[3] |
D. Bian and X. Pu, Global smooth axisymmetic solutions of the Boussinesq equations for magnetohydrodynamics convection, J. Math. Fluid Mech., 22 (2020), Article number: 12, 13pp.
doi: 10.1007/s00021-019-0468-8. |
[4] |
C. Cao and J. Wu,
Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.
doi: 10.1007/s00205-013-0610-3. |
[5] |
D. Chae,
Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.
doi: 10.1016/j.aim.2005.05.001. |
[6] |
D. Chae and J. Lee,
On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671.
doi: 10.1007/s002090100317. |
[7] |
H. Chen, D. Fang and T. Zhang,
Regularity of 3D axisymmetric Navier-Stokes equations, Discrete Contin. Dyn. Syst., 37 (2017), 1923-1939.
doi: 10.3934/dcds.2017081. |
[8] |
Q. Chen, C. Miao and Z. Zhang,
On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.
doi: 10.1007/s00220-008-0545-y. |
[9] |
Q. Chen and Z. Zhang,
Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations, J. Math. Anal. Appl., 331 (2007), 1384-1395.
doi: 10.1016/j.jmaa.2006.09.069. |
[10] |
E. B. Fabes, B. F. Jones and N. M. Rivière,
The initial value problem for the Navier-Stokes equations with data in ${L}^p$, Arch. Rational Mech. Anal., 45 (1972), 222-240.
doi: 10.1007/BF00281533. |
[11] |
Y. Giga,
Solutions for semilinear parabolic equations in ${L}^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[12] |
C. He and Z. Xin,
On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[13] |
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. |
[14] |
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. |
[15] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[16] |
H. Kozono and Y. Taniuchi,
Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys., 214 (2000), 191-200.
doi: 10.1007/s002200000267. |
[17] |
A. Larios, E. Lunasin and E. S. Titi,
Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.
doi: 10.1016/j.jde.2013.07.011. |
[18] |
A. Larios and Y. Pei,
On the local well-posedness and a Prodi-Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion, J. Differential Equations, 263 (2017), 1419-1450.
doi: 10.1016/j.jde.2017.03.024. |
[19] |
Z. Lei,
On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.
doi: 10.1016/j.jde.2015.04.017. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, CRC Press, 2002.
doi: 10.1201/9781420035674.![]() ![]() ![]() |
[21] |
F. Lin, L. Xu and P. Zhang,
Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.
doi: 10.1016/j.jde.2015.06.034. |
[22] |
H. Liu, D. Bian and X. Pu, Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion, Z. Angew. Math. Phys., 70 (2019), Article number: 81, 19pp.
doi: 10.1007/s00033-019-1126-y. |
[23] |
A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, vol. 9 of Courant Lecture Notes in Mathematics, AMS/CIMS, 2003.
doi: 10.1090/cln/009. |
[24] |
C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96 of Appl. Math. Sci., Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
[25] |
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. |
[26] |
G. Mulone and S. Rionero,
Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem, Arch. Ration. Mech. Anal., 166 (2003), 197-218.
doi: 10.1007/s00205-002-0230-9. |
[27] |
X. Pan, Global regularity of solutions for the 3D non-resistive and non-diffusive MHD-Boussinesq system with axisymmetric data, preprint, arXiv: 1911.01550v2. |
[28] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
[29] |
G. Prodi,
Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.
doi: 10.1007/BF02410664. |
[30] |
X. Ren, J. Wu, Z. Xiang and Z. Zhang,
Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.
doi: 10.1016/j.jfa.2014.04.020. |
[31] |
J. Serrin,
On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.
doi: 10.1007/BF00253344. |
[32] |
J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, (1963), 69–98. |
[33] |
M. Struwe,
On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.
doi: 10.1002/cpa.3160410404. |
[34] |
S. Takahashi,
On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math., 69 (1990), 237-254.
doi: 10.1007/BF02567922. |
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