Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 2, pp 161–186 | Cite as

Stability of Rarefaction Wave to the 1-D Piston Problem for the Pressure-Gradient Equations

  • Min DingEmail author
Article
  • 21 Downloads

Abstract

The 1-D piston problem for the pressure gradient equations arising from the flux-splitting of the compressible Euler equations is considered. When the total variations of the initial data and the velocity of the piston are both sufficiently small, the author establishes the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength by employing a modified wave front tracking method.

Keywords

Piston problem Pressure gradient equations Rarefaction wave Wave front tracking method Interaction of waves 

2000 MR Subject Classification

35A01 35L50 35Q35 35R35 76N10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agarwal, R. K. and Halt, D. W., A modified CUSP scheme in wave/particle split from for unstructed grid Euler flow, Frontiers of Computational Fluid Dynamics, Caughey, D. A., Hafez, M. M. (eds.), Chichester, Wiley, New York, 1994.Google Scholar
  2. [2]
    Amadori, D., Initial boundary value problem for nonlinear systems of conservation laws, NoDEA., 4, 1997, 1–42.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bressan, A., Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press Inc., New York, 2000.zbMATHGoogle Scholar
  4. [4]
    Chen, G., Chen, S. and Wang, D., A multidimensional piston problem for the Euler equations for compressible flow, Dis. Cont. Dyn. Sys., 12, 2005, 361–383.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Chen, S., A singular multidimensional piston problem in compressible flow, J. Diff. Eqns., 189, 2003, 292–317.CrossRefzbMATHGoogle Scholar
  6. [6]
    Chen, S., Wang, Z. and Zhang, Y., Global existence of shock front solutions for the axially symmetric piston problem for compressible fluids, J. Hyper. Diff. Eqns., 1, 2004, 51–84.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen, S., Wang, Z. and Zhang, Y., Global existence of shock front solution to axially symmetric piston problem in compressible flow, Z. Angew. Math. Phys., 59, 2008, 434–456.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Applied Mathematical Sciences, 12, Wiley-Interscience, New York, 1948.Google Scholar
  9. [9]
    Ding, M., Kuang, J. and Zhang, Y., Global stability of rarefaction wave to the 1-D piston problem for the Compressible Full Euler Equations, Journal of Mathematical Analysis and Applications, 2, 2017, 1228–1264. DOI: 10.1016/jmaa.2016.11.059.448MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lax, P., Hyperbolic system of conservation laws, II, Comm. Pure. Appl. Math., 10, 1957, 537–566.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Li, Y. and Cao Y., Second order “large particle” difference method, Sciences in China Ser. A., 8, 1985, 1024–1035 (in Chinese).zbMATHGoogle Scholar
  12. [12]
    Smoller, J., Shock Waves and Reaction-Diffusion Equations, Spring-Verlag, New York, 1983.CrossRefzbMATHGoogle Scholar
  13. [13]
    Wang, Z., Local existence of the shock front solution to the axi-symmetrical piston problem in compressible flow, Acta Math. Sin. Engl. Ser., 20, 2004, 589–604.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Wang, Z., Global existence of shock front solution to 1-dimensional piston problem, Chinese Ann. Math. Ser. A, 26(5), 2005, 549–560 (in Chinese).MathSciNetzbMATHGoogle Scholar
  15. [15]
    Zhang, Y., Steady supersonic flow over a bending wall, Nonlinear Analysis: Real World Applications, 12, 2011, 167–189.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Zhou, Y., One-Dimensional Unsteady Fluid Dynamics, 2nd ed., Science Press, Beijing, 1998 (in Chinese).Google Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceWuhan University of TechnologyWuhanChina

Personalised recommendations