American Institute of Mathematical Sciences

Advanced
October  2016, 36(10): 5801-5815. doi: 10.3934/dcds.2016055

On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations

Gaocheng Yue 1, and  Chengkui Zhong 2,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106
2. 

Department of Mathematics, Nanjing University, Nanjing 210093

Received  September 2015 Revised  April 2016 Published  July 2016

The present paper is devoted to the well-posedness issue of solutions to the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations with horizontal dissipation and horizontal magnetic diffusion. By means of anisotropic Littlewood-Paley analysis we prove the global well-posedness of solutions in the anisotropic Sobolev spaces of type $H^{0,s_0}(\mathbb{R}^3)$ with $s_0>\frac1{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} (\|u_n^h(0)\|_{H^{0,s_0}}^2+\|B_n^h(0)\|_{H^{0,s_0}}^2)\exp \Big\{C_1(\|u_0^3\|_{H^{0,s_0}}^4+\|B_0^3\|_{H^{0,s_0}}^4)\Big\}\leq\varepsilon_0, \end{align*} for some sufficiently small constant $\varepsilon_0.$
Keywords: anisotropic Littlewood-Paley theory, MHD equations, Well-posedness, Besov spaces..
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 35Q3.
Citation: Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055
References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion,, J. Differ. Equ., 254 (2013), 2661.  doi: 10.1016/j.jde.2013.01.002.  Google Scholar

[3]

J. Chemin, Localization in Fourier space and Navier-Stokes system Phase space analysis of Partial Differential Equations,, Pubbl. Cert. Ric. Mat. Ennio de Gorg Scuola Norma. Sup. Pisa, I (2004), 53.   Google Scholar

[4]

J. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity,, Math. Model. Numer. Anal., 34 (2000), 315.  doi: 10.1051/m2an:2000143.  Google Scholar

[5]

J. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces,, Adv. Math., 286 (2016), 1.  doi: 10.1016/j.aim.2015.09.004.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations,, Comm. Math. Phys., 275 (2007), 861.  doi: 10.1007/s00220-007-0319-y.  Google Scholar

[7]

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.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

J. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations,, Comm. Math. Phys., 272 (2007), 529.  doi: 10.1007/s00220-007-0236-0.  Google Scholar

[9]

G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations,, Adv. Math., 225 (2010), 1248.  doi: 10.1016/j.aim.2010.03.022.  Google Scholar

[10]

D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces,, Rev. Mat. Iberoamericana, 15 (1999), 1.  doi: 10.4171/RMI/248.  Google Scholar

[11]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity,, Discrete Contin. Dyn. Syst., 25 (2009), 575.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[12]

F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system,, J. Differ. Equ., 259 (2015), 5440.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[13]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case,, Comm. Pure Appl. Math., 67 (2014), 531.  doi: 10.1002/cpa.21506.  Google Scholar

[14]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques,, Rev. Mat. Iberoamericana, 21 (2005), 179.  doi: 10.4171/RMI/420.  Google Scholar

[15]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, Comm. Math. Phys., 307 (2011), 713.  doi: 10.1007/s00220-011-1350-6.  Google Scholar

[16]

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.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[17]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar

[18]

J. Wu, Bounds and new approaches for the 3D MHD equations,, J. Nonlinear Science, 12 (2002), 395.  doi: 10.1007/s00332-002-0486-0.  Google Scholar

[19]

J. Wu, Regularity criteria for the generalized MHD equations,, Commun. Partial Diff. Equ., 33 (2008), 285.  doi: 10.1080/03605300701382530.  Google Scholar

[20]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term,, SIAM J. Math. Anal., 47 (2015), 2630.  doi: 10.1137/140985445.  Google Scholar

[21]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system,, SIAM J. Math. Anal., 47 (2015), 26.  doi: 10.1137/14095515X.  Google Scholar

[22]

T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space,, Comm. Math. Phys., 295 (2010), 877.  doi: 10.1007/s00220-010-1004-0.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion,, J. Differ. Equ., 254 (2013), 2661.  doi: 10.1016/j.jde.2013.01.002.  Google Scholar

[3]

J. Chemin, Localization in Fourier space and Navier-Stokes system Phase space analysis of Partial Differential Equations,, Pubbl. Cert. Ric. Mat. Ennio de Gorg Scuola Norma. Sup. Pisa, I (2004), 53.   Google Scholar

[4]

J. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity,, Math. Model. Numer. Anal., 34 (2000), 315.  doi: 10.1051/m2an:2000143.  Google Scholar

[5]

J. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces,, Adv. Math., 286 (2016), 1.  doi: 10.1016/j.aim.2015.09.004.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations,, Comm. Math. Phys., 275 (2007), 861.  doi: 10.1007/s00220-007-0319-y.  Google Scholar

[7]

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.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

J. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations,, Comm. Math. Phys., 272 (2007), 529.  doi: 10.1007/s00220-007-0236-0.  Google Scholar

[9]

G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations,, Adv. Math., 225 (2010), 1248.  doi: 10.1016/j.aim.2010.03.022.  Google Scholar

[10]

D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces,, Rev. Mat. Iberoamericana, 15 (1999), 1.  doi: 10.4171/RMI/248.  Google Scholar

[11]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity,, Discrete Contin. Dyn. Syst., 25 (2009), 575.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[12]

F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system,, J. Differ. Equ., 259 (2015), 5440.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[13]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case,, Comm. Pure Appl. Math., 67 (2014), 531.  doi: 10.1002/cpa.21506.  Google Scholar

[14]

M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques,, Rev. Mat. Iberoamericana, 21 (2005), 179.  doi: 10.4171/RMI/420.  Google Scholar

[15]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces,, Comm. Math. Phys., 307 (2011), 713.  doi: 10.1007/s00220-011-1350-6.  Google Scholar

[16]

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.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[17]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar

[18]

J. Wu, Bounds and new approaches for the 3D MHD equations,, J. Nonlinear Science, 12 (2002), 395.  doi: 10.1007/s00332-002-0486-0.  Google Scholar

[19]

J. Wu, Regularity criteria for the generalized MHD equations,, Commun. Partial Diff. Equ., 33 (2008), 285.  doi: 10.1080/03605300701382530.  Google Scholar

[20]

J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term,, SIAM J. Math. Anal., 47 (2015), 2630.  doi: 10.1137/140985445.  Google Scholar

[21]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system,, SIAM J. Math. Anal., 47 (2015), 26.  doi: 10.1137/14095515X.  Google Scholar

[22]

T. Zhang, Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space,, Comm. Math. Phys., 295 (2010), 877.  doi: 10.1007/s00220-010-1004-0.  Google Scholar

[1]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865
[2]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[3]

Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1
[4]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[5]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[6]

Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic & Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030
[7]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[8]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[9]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[10]

Tong Li. Well-posedness theory of an inhomogeneous traffic flow model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 401-414. doi: 10.3934/dcdsb.2002.2.401
[11]

Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks & Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601
[12]

Edriss S. Titi, Saber Trabelsi. Global well-posedness of a 3D MHD model in porous media. Journal of Geometric Mechanics, 2019, 11 (4) : 621-637. doi: 10.3934/jgm.2019031
[13]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[14]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555
[15]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[16]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
[17]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813
[18]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605
[19]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253
[20]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences