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

Article

Abstract

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byv tu x = 0,u t +p(v) x = μ(u x/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 +) ∈R 2(v ,u ), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (v r,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 (v r,ur)(x/t) toR + 1 as t → ∞ provided that both the initial perturbations and ¦(v +v ,u +-u t-) are small. The result is given by an elementaryL 2 energy method.

Key words

asymptotic behavior rarefaction wave compressible viscous gas boundary 

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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

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