Asymptotic Stability of the Rarefaction Wave of a One-Dimensional Model System for Compressible Viscous Gas with Boundary

Tao Pan, Hongxia Liu, Kenji Nishihara

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    14 Citations (Scopus)


    This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by vt - ux = 0, ut + p(v)x = μ(ux/v)x on R1 + with boundary. The initial data of (v, u) have constant states (v+, u+) at +∞ and the boundary condition at x = 0 is given only on the velocity u, say u-. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u- < u+, v- is determined as (v+, 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) to R1 + as t → ∞ provided that both the initial perturbations and |(v+ -v-, u+ - u-)| are small. The result is given by an elementary L2 energy method.

    Original languageEnglish
    Pages (from-to)431-441
    Number of pages11
    JournalJapan Journal of Industrial and Applied Mathematics
    Issue number3
    Publication statusPublished - 1999 Oct



    • Asymptotic behavior
    • Boundary
    • Compressible viscous gas
    • Rarefaction wave

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics

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