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.
|ジャーナル||Japan Journal of Industrial and Applied Mathematics|
|出版ステータス||Published - 1999 10|
ASJC Scopus subject areas
- Applied Mathematics