April  2017, 37(4): 2103-2113. doi: 10.3934/dcds.2017090

On the local C1, α solution of ideal magneto-hydrodynamical equations

1. 

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

2. 

School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China

* Corresponding author: Yu-Li Ge, Email:yulixli@126.com

Received  June 2016 Revised  November 2016 Published  December 2016

This paper is devoted to the study of the two-dimensional andthree-dimensional ideal incompressible magneto-hydrodynamic (MHD)equations in which the Faraday law is inviscid. We consider thelocal existence and uniqueness of classical solutions for the MHDsystem in Hölder space when the general initial data belongs to$C^{1,α}(\mathbb{R}^n)$ for $n=2$ and $n=3$.

Citation: Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C1, α solution of ideal magneto-hydrodynamical equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405. 

[2]

C. BardosC. Sulem and P. L. Sulem, Long time dynamics of a conductive fluid in the presence of a strong magnetic field, Trans. Amer. Math. Soc., 305 (1988), 175-191.  doi: 10.1090/S0002-9947-1988-0920153-5.

[3]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[4] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.
[5] H. Cabannes, Theoretical Magneto-Fluid Dynamics, Academic Press, New York, London, 1970. 
[6]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, arXiv: 1605.00439v1 [math. AP], 2 May 2016.

[7]

R. CaflishI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[8]

Q. ChenC. Miao and Z. Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces, Arch. Rational Mech. Anal., 195 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.

[9]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.

[10]

C. Cao and J. Wu, Global regularity theory for the incompressible magnetohydrodynamic type equations, In Lectures on the Analysis of Nonlinear Partial Differential Equations, Morningside Lectures in Mathematics, Edited by Fang-Hua Lin and Ping Zhang, Higher Education Press, Beijing, China, 2 (2012), 19–45.

[11]

T. G. Cowling and D. Phil, Magnetohydrodynamics, the Institute of Physics, 1976.

[12] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.
[13]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. 

[14]

U. FrischA. PouquetP. L. Sulem and M. Meneguzzi, Special issue on two dimensional turbulence, J. Méc. Théor. Appl., 46 (1983), 191-216. 

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Differential Equations of Second Order Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

B. Han, Global strong solution for the density dependent incompressible viscoelastic fluids in the critical $L^p$ framework, Nonlinear Anal., 132 (2016), 337-358.  doi: 10.1016/j.na.2015.11.011.

[17]

B. Han and C. H. Wei, Global well-posedness for the inhomogeneous Navier-Stokes equations with logarithmical hyper-deissipation, Discrete Continuous Dynam. Systems -A, 36 (2016), 6921-6941.  doi: 10.3934/dcds.2016101.

[18]

X. Hu, Z. Lei and F. H. Lin, On magnetohydrodynamics with partial magnetic dissipation near equilibrium, Recent developments in geometry and analysis, 155–164, Adv. Lect. Math. (ALM), 23, Int. Press, Somerville, MA, 2012.

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.

[20]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed, Pergamon, New York, 1984.

[21]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimension, J. Diff. Equations., 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.

[22]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete and Continuous Dynamical Systems., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.

[23]

A. E. Lifschitz, Magnetohydrodynamics and Spectral Theory. Developments in Electromagnetic Theory and Applications, 4, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2561-8.

[24]

F. H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.

[25]

F. H. LinL. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, J. Diff. Equations., 258 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.

[26]

F. H. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580.  doi: 10.1002/cpa.21506.

[27]

F. H. Lin and T. Zhang, Global small solutions to a complex fluid model in three dimensional, Arch. Rational Mech. Anal., 216 (2015), 905-920.  doi: 10.1007/s00205-014-0822-1.

[28]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[29] E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511525087.
[30]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[31]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.  doi: 10.1007/s00332-002-0486-0.

[32]

L. Xu and P. Zhang, Global Small Solutions to Three-Dimensional Incompressible Magnetohydrodynamical System, SIAM J. Math. Anal., 47 (2015), 26-65.  doi: 10.1137/14095515X.

[33]

X. G. Yang and Y. M. Qin, A Beale-Kato-Majda criterion for the 3D viscous magnetohydrodynamic equations, Math. Methods Appl. Sci., 38 (2015), 701-707.  doi: 10.1002/mma.3101.

[34]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv: 1404.5681v2 [math. AP], 23 October, 2014.

show all references

References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405. 

[2]

C. BardosC. Sulem and P. L. Sulem, Long time dynamics of a conductive fluid in the presence of a strong magnetic field, Trans. Amer. Math. Soc., 305 (1988), 175-191.  doi: 10.1090/S0002-9947-1988-0920153-5.

[3]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[4] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.
[5] H. Cabannes, Theoretical Magneto-Fluid Dynamics, Academic Press, New York, London, 1970. 
[6]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, arXiv: 1605.00439v1 [math. AP], 2 May 2016.

[7]

R. CaflishI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[8]

Q. ChenC. Miao and Z. Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces, Arch. Rational Mech. Anal., 195 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.

[9]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.

[10]

C. Cao and J. Wu, Global regularity theory for the incompressible magnetohydrodynamic type equations, In Lectures on the Analysis of Nonlinear Partial Differential Equations, Morningside Lectures in Mathematics, Edited by Fang-Hua Lin and Ping Zhang, Higher Education Press, Beijing, China, 2 (2012), 19–45.

[11]

T. G. Cowling and D. Phil, Magnetohydrodynamics, the Institute of Physics, 1976.

[12] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.
[13]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. 

[14]

U. FrischA. PouquetP. L. Sulem and M. Meneguzzi, Special issue on two dimensional turbulence, J. Méc. Théor. Appl., 46 (1983), 191-216. 

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Differential Equations of Second Order Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

B. Han, Global strong solution for the density dependent incompressible viscoelastic fluids in the critical $L^p$ framework, Nonlinear Anal., 132 (2016), 337-358.  doi: 10.1016/j.na.2015.11.011.

[17]

B. Han and C. H. Wei, Global well-posedness for the inhomogeneous Navier-Stokes equations with logarithmical hyper-deissipation, Discrete Continuous Dynam. Systems -A, 36 (2016), 6921-6941.  doi: 10.3934/dcds.2016101.

[18]

X. Hu, Z. Lei and F. H. Lin, On magnetohydrodynamics with partial magnetic dissipation near equilibrium, Recent developments in geometry and analysis, 155–164, Adv. Lect. Math. (ALM), 23, Int. Press, Somerville, MA, 2012.

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.

[20]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed, Pergamon, New York, 1984.

[21]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimension, J. Diff. Equations., 259 (2015), 3202-3215.  doi: 10.1016/j.jde.2015.04.017.

[22]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete and Continuous Dynamical Systems., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.

[23]

A. E. Lifschitz, Magnetohydrodynamics and Spectral Theory. Developments in Electromagnetic Theory and Applications, 4, Kluwer Academic Publishers Group, Dordrecht, 1989. doi: 10.1007/978-94-009-2561-8.

[24]

F. H. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.

[25]

F. H. LinL. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system, J. Diff. Equations., 258 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.

[26]

F. H. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580.  doi: 10.1002/cpa.21506.

[27]

F. H. Lin and T. Zhang, Global small solutions to a complex fluid model in three dimensional, Arch. Rational Mech. Anal., 216 (2015), 905-920.  doi: 10.1007/s00205-014-0822-1.

[28]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.

[29] E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511525087.
[30]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.

[31]

J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.  doi: 10.1007/s00332-002-0486-0.

[32]

L. Xu and P. Zhang, Global Small Solutions to Three-Dimensional Incompressible Magnetohydrodynamical System, SIAM J. Math. Anal., 47 (2015), 26-65.  doi: 10.1137/14095515X.

[33]

X. G. Yang and Y. M. Qin, A Beale-Kato-Majda criterion for the 3D viscous magnetohydrodynamic equations, Math. Methods Appl. Sci., 38 (2015), 701-707.  doi: 10.1002/mma.3101.

[34]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv: 1404.5681v2 [math. AP], 23 October, 2014.

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