Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

Akitaka Matsumura, Kenji Nishihara

    Research output: Contribution to journalArticle

    112 Citations (Scopus)

    Abstract

    The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation laws ut+f(u)x=μuxx with the initial data u0 which tend to the constant states u± as x→±∞. Stability theorems are obtained in the absence of the convexity of f and in the allowance of s (shock speed)=f′(u±). Moreover, the rate of asymptotics in time is investigated. For the case f′(u+)<s<f′(u-), if the integral of the initial disturbance over (-∞, x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate as t→∞. This rate seems to be almost optimal compared with the rate in the case f=u2/2 for which an explicit form of the solution exists. The rate is also obtained in the case f′(u±=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

    Original languageEnglish
    Pages (from-to)83-96
    Number of pages14
    JournalCommunications in Mathematical Physics
    Volume165
    Issue number1
    DOIs
    Publication statusPublished - 1994 Oct

    Fingerprint

    Viscous Conservation Laws
    Scalar Conservation Laws
    conservation laws
    traveling waves
    Traveling Wave
    Asymptotic Stability
    Shock
    nonlinearity
    Nonlinearity
    scalars
    Stability Theorem
    Energy Method
    Traveling Wave Solutions
    Convexity
    Disturbance
    Decay
    Tend
    shock
    convexity
    energy methods

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Physics and Astronomy(all)
    • Mathematical Physics

    Cite this

    Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. / Matsumura, Akitaka; Nishihara, Kenji.

    In: Communications in Mathematical Physics, Vol. 165, No. 1, 10.1994, p. 83-96.

    Research output: Contribution to journalArticle

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