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

Tai Ping Liu, Akitaka Matsumura, Kenji Nishihara

    Research output: Contribution to journalArticle

    86 Citations (Scopus)

    Abstract

    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
    Volume29
    Issue number2
    Publication statusPublished - 1998 Mar

    Fingerprint

    Rarefaction Wave
    Behavior of Solutions
    Burgers Equation
    Asymptotic Behavior of Solutions
    Far Field
    Generalized Equation
    Shock Waves
    Shock waves
    Global Existence
    Initial-boundary-value Problem
    Boundary value problems
    Dirichlet
    Superposition
    Half line
    Boundary Layer
    Cauchy Problem
    Boundary layers
    Asymptotic Behavior
    Dependent

    Keywords

    • Asymptotic behavior
    • Rarefaction wave
    • Viscous shock wave

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics

    Cite this

    Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves. / Liu, Tai Ping; Matsumura, Akitaka; Nishihara, Kenji.

    In: SIAM Journal on Mathematical Analysis, Vol. 29, No. 2, 03.1998, p. 293-308.

    Research output: Contribution to journalArticle

    Liu, Tai Ping ; Matsumura, Akitaka ; Nishihara, Kenji. / Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves. In: SIAM Journal on Mathematical Analysis. 1998 ; Vol. 29, No. 2. pp. 293-308.
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