We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid--point mass system is governed by the barotropic compressible Navier--Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass, and it gives pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values. As a corollary, it is shown that the velocity $V(t)$ of the point mass satisfies a decay estimate $|V(t)|=O(t^{-3/2})$ --- a faster decay compared to $t^{-1/2}$ known for the motion of a point mass in the viscous Burgers fluid~[J.~L.~V{a}zquez and E.~Zuazua, Comm. Partial Differential Equations \textbf{28} (2003), 1705--1738]. The rate $-3/2$ is essentially related to the compressibility and the nonlinearity. As a consequence, it follows that the point mass is convected only a finite distance as opposed to the viscous Burgers case. The main tool used in the proof is the pointwise estimates of Green's function. It turns out that the understanding of the time-decay properties of the transmitted and reflected waves at the point mass is essential for the proof.

中文翻译：

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一维粘性可压缩流体中点质量的长期行为和解的逐点估计

我们考虑一维粘性可压缩正压流体中点质量的运动。流体点质量系统受正压可压缩 Navier-Stokes 方程和牛顿运动方程控制。我们的主要结果涉及流体和点质量的长期行为，它给出了体积比和流体速度与其平衡值的逐点收敛估计。作为推论，它表明点质量的速度 $V(t)$ 满足衰减估计 $|V(t)|=O(t^{-3/2})$ --- 更快的衰减与 $t^{-1/2}$ 相比，$t^{-1/2}$ 以粘性汉堡流体中点质量的运动而闻名~[J.~L.~V{a}zquez and E.~Zuazua, Comm. 偏微分方程 \textbf{28} (2003), 1705--1738]。比率$-3/2$本质上与可压缩性和非线性有关。因此，与粘性汉堡情况相反，点质量仅对流有限距离。证明中使用的主要工具是格林函数的逐点估计。事实证明，理解点质量处传输和反射波的时间衰减特性对于证明至关重要。