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

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