| Abstract: |
| We consider the motion of a point mass in a one-dimensional viscous barotropic compressible fluid. The point mass is described by Newton`s equations of motion and the fluid by the compressible Navier--Stokes equations. The fluid domain is $\mathbb{R}\backslash \{ h(t) \}$, where $h(t)$ is the location of the point mass. These equations are coupled in two ways through the fluid force in Newton`s equations and the boundary condition for the fluid equations. We show that the velocity $V(t)=h`(t)$ of the point mass decays as $V(t)\sim t^{-3/2}$ as $t\to \infty$. This is proved by extending the pointwise estimate method for the Cauchy problem developed in~[T.-P. Liu and Y. Zeng, Mem. Amer. Math. Soc., \textbf{125}, 1997] to our fluid--structure interaction system. If time allows, we would also present some extensions obtained recently. |
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