Abstract and Applied Analysis

Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness

Fu-zhi Li, Jia-li Yu, Yang-rong Li, and Gan-shan Yang

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Abstract

Based on classical Lie Group method, we construct a class of explicit solutions of two-dimensional ideal incompressible magnetohydrodynamics (MHD) equation by its infinitesimal generator. Via these explicit solutions we study the uniqueness and stability of initial-boundary problem on MHD.

Article information

Source
Abstr. Appl. Anal., Volume 2016, Special Issue (2016), Article ID 8183079, 8 pages.

Dates
Received: 27 June 2016
Accepted: 20 September 2016
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1485313516

Digital Object Identifier
doi:10.1155/2016/8183079

Mathematical Reviews number (MathSciNet)
MR3590296

Zentralblatt MATH identifier
06929390

Citation

Li, Fu-zhi; Yu, Jia-li; Li, Yang-rong; Yang, Gan-shan. Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness. Abstr. Appl. Anal. 2016, Special Issue (2016), Article ID 8183079, 8 pages. doi:10.1155/2016/8183079. https://projecteuclid.org/euclid.aaa/1485313516


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References

  • D. Cordoba and C. Marliani, “Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D,” Communications on Pure and Applied Mathematics, vol. 53, no. 4, pp. 512–524, 2000.
  • D. Biskamp, Nonlinear Magnetohydrodynamics, vol. 1 of Cambridge Monographs on Plasma Physics, Cambridge University Press, Cambridge, Uk, 1993.
  • Q. L. Chen, C. X. Miao, and Z. F. Zhang, “The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations,” Communications in Mathematical Physics, vol. 275, no. 3, pp. 861–872, 2007.
  • J. Wu, “Viscous and inviscid magnetohydrodynamics equations,” Journal d'Analyse Mathématique, vol. 73, pp. 251–265, 1997.
  • C. Cao and J. Wu, “Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion,” https://arxiv.org/abs/0901.2908.
  • C. He and Z. Xin, “Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,” Journal of Functional Analysis, vol. 227, no. 1, pp. 113–152, 2005.
  • V. Vyalov, “Partial regularity of solutions to the magnetohydrodynamic equations,” Journal of Mathematical Sciences (New York), vol. 150, no. 1, pp. 1771–1786, 2008.
  • M. Sermange and R. Temam, “Some mathematical questions related to the MHD equations,” Communications on Pure and Applied Mathematics, vol. 36, no. 5, pp. 635–664, 1983.
  • P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 2nd edition, 1993.
  • S. Yang and C. Hua, “Lie symmetry reductions and exact solutions of a coupled KdV-Burgers equation,” Applied Mathematics and Computation, vol. 234, pp. 579–583, 2014.
  • S. Shen, C. Qu, and Q. Huang, “Lie group classification of the \emphN-th-order nonlinear evolution equations,” Science China Mathematics, vol. 54, no. 12, pp. 2553–2572, 2011.
  • E. Casella, P. Secchi, and P. Trebeschi, “Global classical solutions for MHD system,” Journal of Mathematical Fluid Mechanics, vol. 5, no. 1, pp. 70–91, 2003.
  • X. Liao, The Stability of the Theory, Method and Application, Huazhong University of Science and Technology Press, 2nd edition, 2010.
  • W. Song, H. Li, G. Yang, and G. X. Yuan, “Nonhomogeneous boundary value problem for (\emphI, \emphJ) similar Solutions of incompressible two-dimensional Euler equations,” Journal of Inequalities and Applications, vol. 2014, no. 1, article 277, 2014.
  • M. D. Gunzburger, A. J. Meir, and J. S. Peterson, “On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics,” Mathematics of Computation, vol. 56, no. 194, pp. 523–563, 1991.
  • L.-J. Yang and J.-M. Wang, “Staility of a damped hyperbolic Timoshenko system coupled with a heat equation,” Asian Journal of Control, vol. 16, no. 2, pp. 546–555, 2014.
  • G. Yang, $\delta $-Viscosity solution, blow up solution and global solution of multidimensional Landau-Lifshitz equations [Ph.D. thesis], China Academy of Engineering Physics, Sichuan, China, 2002.
  • V. I. Yudovitch, “Non-stationary flow of an ideal incompressible liquid,” USSR Computational Mathematics and Mathematical Physics, vol. 3, no. 6, pp. 1407–1456, 1963. \endinput