Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions

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. Pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values are shown, and as a corollary, it is proved that the velocity V(t) of the point mass satisfies a decay estimate |V(t)|=O(t^−3/2). The rate −3/2 is optimal and is essentially related to the compressibility and the nonlinearity. 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 plays an essential role in the proof.