| Abstract: |
| We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. We do not impose the zero-mass condition on initial data, which improves the previous results by Matsumura and Mei \cite{MM99} for impermeable case, and by Huang, Matsumura and Shi \cite{HMS03} for inflow case. Moreover, for the inflow case, we remove the assumption $\gamma\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems. |
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