Japan Journal of Industrial and Applied Mathematics

, Volume 16, Issue 3, pp 431–441

Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary


DOI: 10.1007/BF03167367

Cite this article as:
Pan, T., Liu, H. & Nishihara, K. Japan J. Indust. Appl. Math. (1999) 16: 431. doi:10.1007/BF03167367


This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byvtux = 0,ut +p(v)x = μ(ux/v)x onR+1 with boundary. The initial data of (v,u) have constant states (v+,u+) at +∞ and the boundary condition atx = 0 is given only on the velocityu, say u. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, whenu <u+,v is determined as (u+,u+) ∈R2(v,u), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (vr,ur)(x/t) connecting two constant states (v,u) and (v+,u+). Our assertion is that the solution of the original system tends to the restriction of (vr,ur)(x/t) toR+1 as t → ∞ provided that both the initial perturbations and ¦(v+v,u+-ut-) are small. The result is given by an elementaryL2 energy method.

Key words

asymptotic behaviorrarefaction wavecompressible viscous gasboundary

Copyright information

© JJIAM Publishing Committee 1999

Authors and Affiliations

  1. 1.Department of MathematicsGuangxi UniversityNanningP.R. China
  2. 2.Department of MathematicsJinan UniversityGuangzhouP.R. China
  3. 3.School of Political Science and EconomicsWaseda UniversityTokyoJapan