### Abstract

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by v_{t} - u_{x} = 0, u_{t} + p(v)_{x} = μ(u_{x}/v)_{x} on R^{1}
_{+} with boundary. The initial data of (v, u) have constant states (v_{+}, u_{+}) at +∞ and the boundary condition at x = 0 is given only on the velocity u, say u_{-}. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u- < u_{+}, v_{-} is determined as (v_{+}, u_{+}) ∈ R_{2}(v_{-}, u_{-}), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (v^{r}, u^{r})(x/t) connecting two constant states (v_{-}, u_{-}) and (v_{+}, u_{+}). Our assertion is that the solution of the original system tends to the restriction of (v^{r}, u^{r})(x/t) to R^{1}
_{+} as t → ∞ provided that both the initial perturbations and |(v_{+} -v_{-}, u_{+} - u_{-})| are small. The result is given by an elementary L^{2} energy method.

Original language | English |
---|---|

Pages (from-to) | 431-441 |

Number of pages | 11 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 16 |

Issue number | 3 |

Publication status | Published - 1999 Oct |

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### Keywords

- Asymptotic behavior
- Boundary
- Compressible viscous gas
- Rarefaction wave

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Japan Journal of Industrial and Applied Mathematics*,

*16*(3), 431-441.

**Asymptotic Stability of the Rarefaction Wave of a One-Dimensional Model System for Compressible Viscous Gas with Boundary.** / Pan, Tao; Liu, Hongxia; Nishihara, Kenji.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 16, no. 3, pp. 431-441.

}

TY - JOUR

T1 - Asymptotic Stability of the Rarefaction Wave of a One-Dimensional Model System for Compressible Viscous Gas with Boundary

AU - Pan, Tao

AU - Liu, Hongxia

AU - Nishihara, Kenji

PY - 1999/10

Y1 - 1999/10

N2 - This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by vt - ux = 0, ut + p(v)x = μ(ux/v)x on R1 + with boundary. The initial data of (v, u) have constant states (v+, u+) at +∞ and the boundary condition at x = 0 is given only on the velocity u, say u-. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u- < u+, v- is determined as (v+, u+) ∈ R2(v-, u-), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (vr, ur)(x/t) connecting two constant states (v-, u-) and (v+, u+). Our assertion is that the solution of the original system tends to the restriction of (vr, ur)(x/t) to R1 + as t → ∞ provided that both the initial perturbations and |(v+ -v-, u+ - u-)| are small. The result is given by an elementary L2 energy method.

AB - This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed by vt - ux = 0, ut + p(v)x = μ(ux/v)x on R1 + with boundary. The initial data of (v, u) have constant states (v+, u+) at +∞ and the boundary condition at x = 0 is given only on the velocity u, say u-. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u- < u+, v- is determined as (v+, u+) ∈ R2(v-, u-), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (vr, ur)(x/t) connecting two constant states (v-, u-) and (v+, u+). Our assertion is that the solution of the original system tends to the restriction of (vr, ur)(x/t) to R1 + as t → ∞ provided that both the initial perturbations and |(v+ -v-, u+ - u-)| are small. The result is given by an elementary L2 energy method.

KW - Asymptotic behavior

KW - Boundary

KW - Compressible viscous gas

KW - Rarefaction wave

UR - http://www.scopus.com/inward/record.url?scp=0012657637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012657637&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0012657637

VL - 16

SP - 431

EP - 441

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 3

ER -