Table of Contents
ISRN Mathematical Physics
Volume 2012 (2012), Article ID 782306, 30 pages
http://dx.doi.org/10.5402/2012/782306
Research Article

The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System

School of Mathematical Sciences, South China Normal University, Guang Zhou 510631, China

Received 13 October 2011; Accepted 26 October 2011

Academic Editors: U. Kulshreshtha and M. Znojil

Copyright © 2012 Shixiang Ma. 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. Hoff and T. Liu, “The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data,” Indiana University Mathematics Journal, vol. 38, no. 4, pp. 861–915, 1989. View at Publisher · View at Google Scholar
  2. J. Goodman and Z. Xin, “Viscous limits for piecewise smooth solutions to systems of conservation laws,” Archive for Rational Mechanics and Analysis, vol. 121, no. 3, pp. 235–265, 1992. View at Publisher · View at Google Scholar · View at Scopus
  3. S. Ma, “The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers,” Journal of Mathematical Analysis and Applications, vol. 378, no. 1, pp. 268–288, 2011. View at Publisher · View at Google Scholar
  4. H. Wang, “Viscous limits for piecewise smooth solutions of the p-system,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 411–432, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Wang, “Zero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock,” Acta Mathematica Scientia. Series B. English Edition, vol. 28, no. 4, pp. 727–748, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. S. Yu, “Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws,” Archive for Rational Mechanics and Analysis, vol. 146, no. 4, pp. 275–370, 1999. View at Google Scholar · View at Scopus
  7. Z. Xin, “Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases,” Communications on Pure and Applied Mathematics, vol. 46, no. 5, pp. 621–665, 1993. View at Publisher · View at Google Scholar
  8. S. Jiang, G. Ni, and W. Sun, “Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids,” SIAM Journal on Mathematical Analysis, vol. 38, no. 2, pp. 368–384, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. H. Zeng, Asymptotic behavior of solutions to fluid dynamical equations, Doctoral thesis, The Chinese University of Hong Kong.
  10. A. Szepessy and K. Zumbrun, “Stability of rarefaction waves in viscous media,” Archive for Rational Mechanics and Analysis, vol. 133, no. 3, pp. 249–298, 1996. View at Google Scholar · View at Scopus
  11. C. Mascia and K. Zumbrun, “Stability of large-amplitude shock profiles of general relaxation systems,” SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 889–913, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Mascia and K. Zumbrun, “Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems,” Archive for Rational Mechanics and Analysis, vol. 172, no. 1, pp. 93–131, 2004. View at Google Scholar
  13. K. Zumbrun and P. Howard, “Pointwise semigroup methods and stability of viscous shock waves,” Indiana University Mathematics Journal, vol. 47, no. 3, pp. 741–871, 1998. View at Google Scholar · View at Scopus
  14. O. Guès, G. Métivier, M. Williams, and K. Zumbrun, “Existence and stability of multidimensional shock fronts in the vanishing viscosity limit,” Archive for Rational Mechanics and Analysis, vol. 175, no. 2, pp. 151–244, 2005. View at Publisher · View at Google Scholar · View at Scopus
  15. F. Huang, Y. Wang, and T. Yang, “Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity,” Kinetic and Related Models, vol. 3, no. 4, pp. 685–728, 2010. View at Publisher · View at Google Scholar
  16. S. Ma, “Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,” Journal of Differential Equations, vol. 248, no. 1, pp. 95–110, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. S. Ma, “Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 1033–1043, 2012. View at Publisher · View at Google Scholar
  18. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1973.
  19. H. O. Kreiss, “Initial boundary value problems for hyperbolic systems,” Communications on Pure and Applied Mathematics, vol. 23, pp. 277–298, 1970. View at Google Scholar
  20. J. Rauch, “L2 is a continuable initial condition for Kreiss' mixed problems,” Communications on Pure and Applied Mathematics, vol. 25, pp. 265–285, 1972. View at Google Scholar
  21. S. Kawashima, Systems of hyperbolic-parabolic composite type, with appli-cations to the equations of magneto-hydrodynamics, Doctoral thesis, Kyoto University, 1983.