Refined pointwise estimates for solutions to the 1D barotropic compressible Navier--Stokes equations: An application to the long-time behavior of a point mass

We study the long-time behavior of a point mass moving in a one-dimensional viscous compressible fluid. Previously, we showed that the velocity of the point mass $V(t)$ satisfies a decay estimate $V(t)=O(t^{-3\mathrm{/}2})$~[K. Koike, J. Differential Equations $\mathbf{\text{271}}$ (2021) 356–413]. This result was obtained as a corollary to pointwise estimates of solutions to a free boundary problem of barotropic compressible Navier–Stokes equations. In this paper, we give a simple necessary and sufficient condition on the initial data for the decay estimate $V(t)=O(t^{-3\mathrm{/}2})$ to be optimal. This is achieved by refining the pointwise estimates previously obtained: we make use of \textit{inter-diffusion waves} that, together with the classical $\mathit{\text{diffusion}}\mathit{\text{waves}}$, give an improved approximation of the fluid behavior around the point mass; this then leads to a sharper understanding of the long-time behavior of the point mass.