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Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion
School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510520, China |
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.
References:
| [1] |
D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511599965.
|
| [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. |
| [3] |
P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511626333.
|
| [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. |
| [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. |
| [6] |
L. Laudau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, OxfordLondon-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
W. Tan, On the global existence for a coupled parabolic-elliptic equations in three dimensions, submitted. |
| [12] |
J. Wu,
Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.
doi: 10.1016/j.jde.2003.07.007. |
show all references
References:
| [1] |
D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511599965.
|
| [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. |
| [3] |
P. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511626333.
|
| [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. |
| [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. |
| [6] |
L. Laudau and E. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, OxfordLondon-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
W. Tan, On the global existence for a coupled parabolic-elliptic equations in three dimensions, submitted. |
| [12] |
J. Wu,
Generalized MHD equations, J. Differential Equations., 195 (2003), 284-312.
doi: 10.1016/j.jde.2003.07.007. |
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