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-<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-<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.
|ジャーナル||Quarterly of Applied Mathematics|
|出版ステータス||Published - 2000 3月|
ASJC Scopus subject areas