### Abstract

The initial-boundary value problem on the half-line R_{+} = (0, ∞) for a system of barotropic viscous flow v_{t}-u_{x} = 0, u_{t}+p(v)_{x} = μ(^{u}x/v)_{x} is investigated, where the pressure p(v) = v^{-γ} (γ≥1) for the specific volume v>0. Note that the boundary value at x = 0 is given only for the velocity u, say u_{-}, and that the initial data (v_{0}, u_{0})(x) have the constant states (v_{+}, u_{+}) at x = +∞ with v_{0}(x)>0, v_{+}>0. If u_{-}<u_{+}, then there is a unique v_{-} such that (v_{+},u_{+})∈ R_{2}(v_{-},u_{-}) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v_{2}
^{R}, u_{2}
^{R})(x/t) connecting (v_{-},u_{-}) with (v_{+},u_{+}). Our assertion is that, if u_{-}<u_{+}, then there exists a global solution (v, u)(t,x) in C^{0}([0, ∞); H^{1}(R_{+})), which tends to the 2-rarefaction wave (v_{2}
^{R}, u_{2}
^{R})(x/t)|_{x≥0} as t→∞ in the maximum norm, with no smallness condition on |u_{+}-u_{-}| and ∥(v_{0}-v_{+}, u_{0}-u_{+})∥_{H(1)}, nor restriction on γ(≥1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L^{2}-energy method.

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

Pages (from-to) | 69-83 |

Number of pages | 15 |

Journal | Quarterly of Applied Mathematics |

Volume | 58 |

Issue number | 1 |

Publication status | Published - 2000 Mar |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Quarterly of Applied Mathematics*,

*58*(1), 69-83.

**Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect.** / Matsumura, Akitaka; Nishihara, Kenji.

Research output: Contribution to journal › Article

*Quarterly of Applied Mathematics*, vol. 58, no. 1, pp. 69-83.

}

TY - JOUR

T1 - Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect

AU - Matsumura, Akitaka

AU - Nishihara, Kenji

PY - 2000/3

Y1 - 2000/3

N2 - The initial-boundary value problem on the half-line R+ = (0, ∞) for a system of barotropic viscous flow vt-ux = 0, ut+p(v)x = μ(ux/v)x is investigated, where the pressure p(v) = v-γ (γ≥1) for the specific volume v>0. Note that the boundary value at x = 0 is given only for the velocity u, say u-, and that the initial data (v0, u0)(x) have the constant states (v+, u+) at x = +∞ with v0(x)>0, v+>0. If u-+, then there is a unique v- such that (v+,u+)∈ R2(v-,u-) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v2 R, u2 R)(x/t) connecting (v-,u-) with (v+,u+). Our assertion is that, if u-+, then there exists a global solution (v, u)(t,x) in C0([0, ∞); H1(R+)), which tends to the 2-rarefaction wave (v2 R, u2 R)(x/t)|x≥0 as t→∞ in the maximum norm, with no smallness condition on |u+-u-| and ∥(v0-v+, u0-u+)∥H(1), nor restriction on γ(≥1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L2-energy method.

AB - The initial-boundary value problem on the half-line R+ = (0, ∞) for a system of barotropic viscous flow vt-ux = 0, ut+p(v)x = μ(ux/v)x is investigated, where the pressure p(v) = v-γ (γ≥1) for the specific volume v>0. Note that the boundary value at x = 0 is given only for the velocity u, say u-, and that the initial data (v0, u0)(x) have the constant states (v+, u+) at x = +∞ with v0(x)>0, v+>0. If u-+, then there is a unique v- such that (v+,u+)∈ R2(v-,u-) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v2 R, u2 R)(x/t) connecting (v-,u-) with (v+,u+). Our assertion is that, if u-+, then there exists a global solution (v, u)(t,x) in C0([0, ∞); H1(R+)), which tends to the 2-rarefaction wave (v2 R, u2 R)(x/t)|x≥0 as t→∞ in the maximum norm, with no smallness condition on |u+-u-| and ∥(v0-v+, u0-u+)∥H(1), nor restriction on γ(≥1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L2-energy method.

UR - http://www.scopus.com/inward/record.url?scp=0033886978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033886978&partnerID=8YFLogxK

M3 - Article

VL - 58

SP - 69

EP - 83

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 1

ER -