Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations

Kenji Nishihara, Tong Yang, Huijiang Zhao

    Research output: Contribution to journalArticle

    81 Citations (Scopus)

    Abstract

    This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (VR,U R,SR)(t, x). If the initial data (v0,u 0,s0)(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (VR, UR, SR)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent γ is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

    Original languageEnglish
    Pages (from-to)1561-1597
    Number of pages37
    JournalSIAM Journal on Mathematical Analysis
    Volume35
    Issue number6
    DOIs
    Publication statusPublished - 2004

    Fingerprint

    Rarefaction Wave
    Compressible Navier-Stokes Equations
    Nonlinear Stability
    Navier Stokes equations
    Global Stability
    Compressible Euler Equations
    Euler equations
    Cauchy Problem
    Tend
    Global Smooth Solution
    Ideal Gas
    Gases
    Small Perturbations
    Japan
    Strictly
    Asymptotic Behavior
    Exponent
    Infinity
    Perturbation

    Keywords

    • Compressible Navier-Stokes equations
    • Global stability
    • Strong rarefaction waves

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics
    • Numerical Analysis

    Cite this

    Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. / Nishihara, Kenji; Yang, Tong; Zhao, Huijiang.

    In: SIAM Journal on Mathematical Analysis, Vol. 35, No. 6, 2004, p. 1561-1597.

    Research output: Contribution to journalArticle

    Nishihara, Kenji ; Yang, Tong ; Zhao, Huijiang. / Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. In: SIAM Journal on Mathematical Analysis. 2004 ; Vol. 35, No. 6. pp. 1561-1597.
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