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 $V(t)=O(t^{-3/2})$~[K. Koike, J. Differential Equations \textbf{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 $V(t)=O(t^{-3/2})$ 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 \textit{diffusion 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.Comment: Ver 1: 54 pages. Any comments on the paper (reports of typos or logical errors, suggestions to increase readability) are welcome. Ver 2: 59 pages, 1 figure. NOTE: The title of the paper is slightly changed. We largely rewritten the introduction and changed the organization of the proof. The mathematical content of the paper is essentially unchanged. Some minor errors are fixe