| Abstract: |
| We study an initial-boundary value problem for the 3D barotropic Navier-Stokes equations. More specifically, we consider the outflow problem on the spatial domain $\mathbb{R}_+ \times \mathbb{T}^2$, where the far-field condition is given by two different constant states. In this paper, our main result is the orbital stability of the viscous shock wave in the subsonic case, assuming that both the shock amplitude and the initial perturbation are small.
The core of our proof consists of two main steps: obtaining a priori zeroth-order estimates and higher-order estimates. In the first step, we utilize the method of $a$-contraction with shifts, originally introduced by Kang and Vasseur in 2017, which is highly effective for controlling the effects of the viscous shock. In the second step, we employ the classical energy method to derive sharp estimates for controlling the boundary terms. |
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