### 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 |

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*Quarterly of Applied Mathematics*,

*58*(1), 69-83.