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June  2021, 26(6): 3271-3278. doi: 10.3934/dcdsb.2020227

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

Yingdan Ji and  Wen Tan ,

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.

Keywords: 3D Stokes-Magneto equations, Fractional magnetic diffusion, Global well-posedness.
Mathematics Subject Classification: Primary: 35M11, 35Q35, 76D03, 76W05.
Citation: Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227
References:
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J. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.  Google Scholar

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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

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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

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D. McCormick, J. 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

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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

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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

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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

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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. Chemin, D. McCormick, J. 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. Fefferman, D. McCormick, J. 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. McCormick, J. 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

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