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Abstract and Applied Analysis
Volume 2014, Article ID 728715, 8 pages
http://dx.doi.org/10.1155/2014/728715
Research Article

Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations

1College of Teacher Education, Quzhou University, Quzhou 324000, China
2College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received 5 April 2014; Accepted 11 June 2014; Published 30 June 2014

Academic Editor: Igor Boglaev

Copyright © 2014 Jian Liu and Ruxu Lian. 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. J. F. Gerbeau and B. Perthame, “Derivation of viscous Saint-Venant system for laminar shallow water, numerical validation,” Discrete and Continuous Dynamical Systems B, vol. 1, no. 1, pp. 89–102, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. F. Marche, “Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects,” European Journal of Mechanics—B/Fluids, vol. 26, no. 1, pp. 49–63, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. D. Bresch and B. Desjardins, “Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,” Communications in Mathematical Physics, vol. 238, no. 1-2, pp. 211–223, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. D. Bresch and B. Desjardins, “On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models,” Journal de Mathématiques Pures et Appliquées, vol. 86, no. 4, pp. 362–368, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. D. Fang and T. Zhang, “Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data,” Journal of Differential Equations, vol. 222, no. 1, pp. 63–94, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A. Mellet and A. Vasseur, “Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations,” SIAM Journal on Mathematical Analysis, vol. 39, no. 4, pp. 1344–1365, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. H.-L. Li, J. Li, and Z.-P. Xin, “Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations,” Communications in Mathematical Physics, vol. 281, no. 2, pp. 401–444, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. R. Danchin, “Global existence in critical spaces for compressible Navier-Stokes equations,” Inventiones Mathematicae, vol. 141, no. 3, pp. 579–614, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. E. Feireisl, A. Novotný, and H. Petzeltová, “On the existence of globally defined weak solutions to the Navier-Stokes equations,” Journal of Mathematical Fluid Mechanics, vol. 3, no. 4, pp. 358–392, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  10. X. D. Huang, J. Li, and Z. P. Xin, “Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 65, no. 4, pp. 549–585, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Jiang and P. Zhang, “On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,” Communications in Mathematical Physics, vol. 215, no. 3, pp. 559–581, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2 of Compressible Models, Oxford University Press, New York, NY, USA, 1998. View at MathSciNet
  13. R. X. Lian, J. Liu, H. L. Li, and L. Xiao, “Cauchy problem for the one-dimensional compressible Navier-Stokes equations,” Acta Mathematica Scientia B: English Edition, vol. 32, no. 1, pp. 315–324, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Frid and V. Shelukhin, “Boundary layers for the Navier-Stokes equations of compressible fluids,” Communications in Mathematical Physics, vol. 208, no. 2, pp. 309–330, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. Frid and V. V. Shelukhin, “Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry,” SIAM Journal on Mathematical Analysis, vol. 31, no. 5, pp. 1144–1156, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Fan and S. Jiang, “Zero shear viscosity limit for the Navier-Stokes equations of compressible isentropic fluids with cylindric symmetry,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 65, no. 1, pp. 35–52, 2007. View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. Jiang and J. W. Zhang, “Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry,” SIAM Journal on Mathematical Analysis, vol. 41, no. 1, pp. 237–268, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. L. Yao, T. Zhang, and C. Zhu, “Boundary layers for compressible Navier-Stokes equations with density-dependent viscosity and cylindrical symmetry,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire Articles, vol. 28, no. 5, pp. 677–709, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Z. Guo, H.-L. Li, and Z. Xin, “Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations,” Communications in Mathematical Physics, vol. 309, no. 2, pp. 371–412, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus