One component regularity criteria for the axially symmetric MHD-Boussinesq system

Previous Article
"Large" strange attractors in the unfolding of a heteroclinic attractor
DCDS Home
This Issue
Next Article
Refinements of topological invariants of flows

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

One component regularity criteria for the axially symmetric MHD-Boussinesq system

Zijin Li ^1,, and Xinghong Pan ^2,

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.

Keywords: magnetohydrodynamics, Boussinesq, axially symmetric, regularity criteria, velocity.

Mathematics Subject Classification: 35Q35, 76D03.

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. 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.

[1]	Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064
[2]	Wojciech M. Zajączkowski. Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1447-1482. doi: 10.3934/cpaa.2019070
[3]	Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110
[4]	Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012
[5]	Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure and Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[6]	Jae-Myoung Kim. Local regularity of the magnetohydrodynamics equations near the curved boundary. Communications on Pure and Applied Analysis, 2016, 15 (2) : 507-517. doi: 10.3934/cpaa.2016.15.507
[7]	Anthony Suen. Global regularity for the 3D compressible magnetohydrodynamics with general pressure. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2927-2943. doi: 10.3934/dcds.2022004
[8]	Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure and Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309
[9]	Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873
[10]	Dongho Chae, Shangkun Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5267-5285. doi: 10.3934/dcds.2016031
[11]	Xumin Gu. Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition. Communications on Pure and Applied Analysis, 2019, 18 (2) : 569-602. doi: 10.3934/cpaa.2019029
[12]	Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55
[13]	Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure and Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839
[14]	Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[15]	Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2299-2337. doi: 10.3934/dcdsb.2018101
[16]	Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237
[17]	Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
[18]	Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695
[19]	Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
[20]	Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Zijin Li Xinghong Pan
on Google Scholar
Zijin Li Xinghong Pan

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Zijin Li Xinghong Pan
on Google Scholar
Zijin Li Xinghong Pan

[Back to Top]