Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary

Tao Pan, Hongxia Liu, Kenji Nishihara

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    Abstract

    We consider the initial-boundary value problem for a one-dimensional compressible viscous gas with free boundary, which is modeled in the Eulerian coordinate as (IBVP) { ρ + (ρu)x = 0, x > x(t), t > 0, (ρu)t + (ρu2 + p)x = μuxx, x > x(t), t > 0, (p - μux)|x=x(t) = p0, dx(t)/dt = u(x(t),t), t ≥ 0, (ρ, u)|t=0 = (ρ0, u0)(x), x ≥ x(0). Here, ρ(> 0) is the density, u is the velocity, p = p(ρ) = ργ (γ ≥ 1: the adiabatic constant) is the pressure, and μ(> 0) is the viscosity constant. At the boundary the flow is attached to the atmosphere with pressure p0 (> 0) and the boundary condition is derived by the balance law. The initial data have constant states (ρ+, u+) at x = ∞. The flow has no vacuum state so that ρ0(x) > 0 and ρ+ > 0 are assumed. Our main purpose is to investigate the asymptotic behaviors of solutions for (IBVP), which are closely related to those for the corresponding Cauchy problem and hence the corresponding Riemann problem. Depending on p0 and the endstates (ρ+, u+), the solutions are shown to tend to the outgoing rarefaction wave or the outgoing viscous shock wave as t tends to infinity. The proof is given under the weakness assumption of the waves. The analysis will be done by changing (IBVP) into the problem in the Lagrangian coordinate.

    Original languageEnglish
    Pages (from-to)273-291
    Number of pages19
    JournalSIAM Journal on Mathematical Analysis
    Volume34
    Issue number2
    DOIs
    Publication statusPublished - 2003

    Fingerprint

    Free Boundary
    Asymptotic Behavior
    Cauchy Problem
    Gases
    Shock waves
    Tend
    Lagrangian Coordinates
    Boundary value problems
    Rarefaction Wave
    Balance Laws
    Boundary conditions
    Vacuum
    Viscosity
    Asymptotic Behavior of Solutions
    Shock Waves
    Initial-boundary-value Problem
    Atmosphere
    Infinity
    Gas

    Keywords

    • Asymptotic behavior
    • Free boundary
    • One-dimensional compressible viscous gas
    • Rarefaction wave
    • Viscous shock wave

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics

    Cite this

    Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary. / Pan, Tao; Liu, Hongxia; Nishihara, Kenji.

    In: SIAM Journal on Mathematical Analysis, Vol. 34, No. 2, 2003, p. 273-291.

    Research output: Contribution to journalArticle

    Pan, Tao ; Liu, Hongxia ; Nishihara, Kenji. / Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary. In: SIAM Journal on Mathematical Analysis. 2003 ; Vol. 34, No. 2. pp. 273-291.
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    N2 - We consider the initial-boundary value problem for a one-dimensional compressible viscous gas with free boundary, which is modeled in the Eulerian coordinate as (IBVP) { ρ + (ρu)x = 0, x > x(t), t > 0, (ρu)t + (ρu2 + p)x = μuxx, x > x(t), t > 0, (p - μux)|x=x(t) = p0, dx(t)/dt = u(x(t),t), t ≥ 0, (ρ, u)|t=0 = (ρ0, u0)(x), x ≥ x(0). Here, ρ(> 0) is the density, u is the velocity, p = p(ρ) = ργ (γ ≥ 1: the adiabatic constant) is the pressure, and μ(> 0) is the viscosity constant. At the boundary the flow is attached to the atmosphere with pressure p0 (> 0) and the boundary condition is derived by the balance law. The initial data have constant states (ρ+, u+) at x = ∞. The flow has no vacuum state so that ρ0(x) > 0 and ρ+ > 0 are assumed. Our main purpose is to investigate the asymptotic behaviors of solutions for (IBVP), which are closely related to those for the corresponding Cauchy problem and hence the corresponding Riemann problem. Depending on p0 and the endstates (ρ+, u+), the solutions are shown to tend to the outgoing rarefaction wave or the outgoing viscous shock wave as t tends to infinity. The proof is given under the weakness assumption of the waves. The analysis will be done by changing (IBVP) into the problem in the Lagrangian coordinate.

    AB - We consider the initial-boundary value problem for a one-dimensional compressible viscous gas with free boundary, which is modeled in the Eulerian coordinate as (IBVP) { ρ + (ρu)x = 0, x > x(t), t > 0, (ρu)t + (ρu2 + p)x = μuxx, x > x(t), t > 0, (p - μux)|x=x(t) = p0, dx(t)/dt = u(x(t),t), t ≥ 0, (ρ, u)|t=0 = (ρ0, u0)(x), x ≥ x(0). Here, ρ(> 0) is the density, u is the velocity, p = p(ρ) = ργ (γ ≥ 1: the adiabatic constant) is the pressure, and μ(> 0) is the viscosity constant. At the boundary the flow is attached to the atmosphere with pressure p0 (> 0) and the boundary condition is derived by the balance law. The initial data have constant states (ρ+, u+) at x = ∞. The flow has no vacuum state so that ρ0(x) > 0 and ρ+ > 0 are assumed. Our main purpose is to investigate the asymptotic behaviors of solutions for (IBVP), which are closely related to those for the corresponding Cauchy problem and hence the corresponding Riemann problem. Depending on p0 and the endstates (ρ+, u+), the solutions are shown to tend to the outgoing rarefaction wave or the outgoing viscous shock wave as t tends to infinity. The proof is given under the weakness assumption of the waves. The analysis will be done by changing (IBVP) into the problem in the Lagrangian coordinate.

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