• Previous Article
    Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum
  • DCDS-B Home
  • This Issue
  • Next Article
    Random attractors for 2D stochastic micropolar fluid flows on unbounded domains
doi: 10.3934/dcdsb.2020227

Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion

School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510520, China

* Corresponding author: Wen Tan

Received  April 2020 Published  July 2020

Fund Project: The first author is supported by NSFC grant 11701099, The second author is supported by NSFC grant 11871346

This paper is devoted to the global well-posedness of a three-dimensional Stokes-Magneto equations with fractional magnetic diffusion. It is proved that the equations admit a unique global-in-time strong solution for arbitrary initial data when the fractional index $ \alpha\ge\frac32 $. This result might have a potential application in the theory of magnetic relaxtion.

Citation: Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020227
References:
[1] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.  Google Scholar
[2]

J. CheminD. McCormickJ. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.  Google Scholar

[3] P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[4]

C. FeffermanD. McCormickJ. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267 (2014), 1035-1056.  doi: 10.1016/j.jfa.2014.03.021.  Google Scholar

[5]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, 2nd ed. Springer, Berlin, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[6]

L. Laudau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, OxfordLondon-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[7]

D. McCormickJ. Robinson and J. Rodrigo, Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation, Arch. Ration. Mech. Anal., 214 (2014), 503-523.  doi: 10.1007/s00205-014-0760-y.  Google Scholar

[8]

H. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals, J. Fluid Mech., 159 (1985), 359-378.  doi: 10.1017/S0022112085003251.  Google Scholar

[9]

H. Moffatt, Relaxation routes to steady Euler flows of complex topology (2009), http://www2.warwick.ac.uk/fac/sci/maths/research/miraw/days/t3_d5_he/keith.pdf.Slides of talk given during MIRaW Day, "Weak Solutions of the 3D Euler Equations", University of Warwick, 8th June 2009. Google Scholar

[10]

J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516.  doi: 10.1142/S0219199799000183.  Google Scholar

[11]

W. Tan, On the global existence for a coupled parabolic-elliptic equations in three dimensions, submitted. Google Scholar

[12]

J. Wu, Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.  Google Scholar

show all references

References:
[1] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.  Google Scholar
[2]

J. CheminD. McCormickJ. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.  Google Scholar

[3] P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[4]

C. FeffermanD. McCormickJ. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267 (2014), 1035-1056.  doi: 10.1016/j.jfa.2014.03.021.  Google Scholar

[5]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, 2nd ed. Springer, Berlin, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[6]

L. Laudau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, OxfordLondon-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[7]

D. McCormickJ. Robinson and J. Rodrigo, Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation, Arch. Ration. Mech. Anal., 214 (2014), 503-523.  doi: 10.1007/s00205-014-0760-y.  Google Scholar

[8]

H. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals, J. Fluid Mech., 159 (1985), 359-378.  doi: 10.1017/S0022112085003251.  Google Scholar

[9]

H. Moffatt, Relaxation routes to steady Euler flows of complex topology (2009), http://www2.warwick.ac.uk/fac/sci/maths/research/miraw/days/t3_d5_he/keith.pdf.Slides of talk given during MIRaW Day, "Weak Solutions of the 3D Euler Equations", University of Warwick, 8th June 2009. Google Scholar

[10]

J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516.  doi: 10.1142/S0219199799000183.  Google Scholar

[11]

W. Tan, On the global existence for a coupled parabolic-elliptic equations in three dimensions, submitted. Google Scholar

[12]

J. Wu, Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.  doi: 10.1016/j.jde.2003.07.007.  Google Scholar

[1]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020142

[2]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[3]

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

[4]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[5]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087

[6]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[7]

Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301

[8]

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

[9]

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

[10]

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

[11]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371

[12]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

[13]

Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072

[14]

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

[15]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[16]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[17]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845

[18]

Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113

[19]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[20]

Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59

2019 Impact Factor: 1.27

Article outline

[Back to Top]