Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2016, Article ID 8183079, 8 pages
http://dx.doi.org/10.1155/2016/8183079
Research Article

Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness

1School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
3School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan 650500, China

Received 27 June 2016; Accepted 20 September 2016

Academic Editor: Bingwen Liu

Copyright © 2016 Fu-zhi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. D. Biskamp, Nonlinear Magnetohydrodynamics, vol. 1 of Cambridge Monographs on Plasma Physics, Cambridge University Press, Cambridge, Uk, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. Wu, “Viscous and inviscid magnetohydrodynamics equations,” Journal d'Analyse Mathématique, vol. 73, pp. 251–265, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  5. 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.
  6. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. V. Vyalov, “Partial regularity of solutions to the magnetohydrodynamic equations,” Journal of Mathematical Sciences (New York), vol. 150, no. 1, pp. 1771–1786, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  9. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 2nd edition, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Shen, C. Qu, and Q. Huang, “Lie group classification of the N-th-order nonlinear evolution equations,” Science China Mathematics, vol. 54, no. 12, pp. 2553–2572, 2011. View at Publisher · View at Google Scholar
  12. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  13. X. Liao, The Stability of the Theory, Method and Application, Huazhong University of Science and Technology Press, 2nd edition, 2010.
  14. W. Song, H. Li, G. Yang, and G. X. Yuan, “Nonhomogeneous boundary value problem for (I, J) similar Solutions of incompressible two-dimensional Euler equations,” Journal of Inequalities and Applications, vol. 2014, no. 1, article 277, 2014. View at Publisher · View at Google Scholar · View at Scopus
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  16. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. G. Yang, δ-Viscosity solution, blow up solution and global solution of multidimensional Landau-Lifshitz equations [Ph.D. thesis], China Academy of Engineering Physics, Sichuan, China, 2002.
  18. 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. View at Publisher · View at Google Scholar