### 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} + (ρu^{2} + p)_{x} = μu_{xx}, x > x(t), t > 0, (p - μu_{x})|_{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) = ρ^{γ} (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 p_{0} (> 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 ρ_{0}(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 p_{0} 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 language | English |
---|---|

Pages (from-to) | 273-291 |

Number of pages | 19 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 34 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 |

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### 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

*SIAM Journal on Mathematical Analysis*,

*34*(2), 273-291. https://doi.org/10.1137/S0036141001385745

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

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 34, no. 2, pp. 273-291. https://doi.org/10.1137/S0036141001385745

}

TY - JOUR

T1 - Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary

AU - Pan, Tao

AU - Liu, Hongxia

AU - Nishihara, Kenji

PY - 2003

Y1 - 2003

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.

KW - Asymptotic behavior

KW - Free boundary

KW - One-dimensional compressible viscous gas

KW - Rarefaction wave

KW - Viscous shock wave

UR - http://www.scopus.com/inward/record.url?scp=0037239099&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037239099&partnerID=8YFLogxK

U2 - 10.1137/S0036141001385745

DO - 10.1137/S0036141001385745

M3 - Article

AN - SCOPUS:0037239099

VL - 34

SP - 273

EP - 291

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -