May  2022, 42(5): 2333-2353. doi: 10.3934/dcds.2021192

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

*Corresponding author: Zijin Li

Received  January 2021 Revised  November 2021 Published  May 2022 Early access  December 2021

Fund Project: Z. Li is supported by Natural Science Foundation of Jiangsu Province (No. BK20200803), National Natural Science Foundation of China (No. 12001285) and the Startup Foundation for Introducing Talent of NUIST (No. 2019r033). X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. BK20180414) and National Natural Science Foundation of China (No. 11801268; No. 12031006)

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.

Citation: Zijin Li, Xinghong Pan. One component regularity criteria for the axially symmetric MHD-Boussinesq system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2333-2353. doi: 10.3934/dcds.2021192
References:
[1]

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.

[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. ChenD. 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. ChenC. 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. FabesB. 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. LariosE. 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. LinL. 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. RenJ. WuZ. 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. 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.

[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. ChenD. 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. ChenC. 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. FabesB. 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. LariosE. 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. LinL. 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. RenJ. WuZ. 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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