Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves

Tai Ping Liu, Akitaka Matsumura, Kenji Nishihara

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    93 Citations (Scopus)


    We investigate the asymptotic behaviors of solutions of the initial-boundary value problem to the generalized Burgers equation ut + f(u)x = uxx on the half-line with the conditions u(0, t) = u-, u(∞, t) = u+, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic state. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f′(u±) of the boundary state u- = u(0) and the far field state u+ = u(∞). In all cases both global existence of the solution and the asymptotic behavior are shown without smallness conditions. New wave phenomena are observed. For instance, when f′(u-) < 0 < f′(u+), the solution behaves as the superposition of (a part of) a viscous shock wave as boundary layer and a rarefaction wave propagating away from the boundary.

    Original languageEnglish
    Pages (from-to)293-308
    Number of pages16
    JournalSIAM Journal on Mathematical Analysis
    Issue number2
    Publication statusPublished - 1998 Mar



    • Asymptotic behavior
    • Rarefaction wave
    • Viscous shock wave

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics

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