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Global asymptotics toward the rarefaction wave for solutions of viscous \(p\)-system with boundary effect. (English) Zbl 1157.35318

Summary: The initial-boundary value problem on the half-line \(R_+=(0,\infty)\) for a system of barotropic viscous flow \(v_t-u_x=0, u_t+p(v)_x=\mu(u_x/v)_x\) is investigated, where the pressure \(p(v)=v^{-\gamma} (\gamma\geq1)\) 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 \(x=+\infty\) with \(v_0(x)>0, v_+>0\). If \(u_-<u_+\), then there is a unique \(v_-\) such that \((v_+,u_+)\in R_2(v_-,u_-)\) (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 \(C^0([0,\infty);H^1(R_+))\), which tends to the 2-rarefaction wave \((v_2^R,u_2^R)(x/t)|_{x\geq 0}\) as \(t\to\infty\) in the maximum norm, with no smallness condition on \(|u_+-u_-|\) and \(\|(v_0-v_+,u_0-u_+)\|_{H^1}\), nor restriction on \(\gamma (\geq1)\). A similar result for the corresponding Cauchy problem is also obtained. The proofs are given by an elementary \(L^2\)-energy method.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
76L05 Shock waves and blast waves in fluid mechanics
35L50 Initial-boundary value problems for first-order hyperbolic systems
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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