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 & Continuous Dynamical Systems - A, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

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

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

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

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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