Long-time behavior of several point particles in a 1D viscous compressible fluid

We study the long-time behavior of several point particles in a 1D viscous compressible fluid. It is shown that the velocities of the point particles all obey the power law t-3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-3/2}$$\end{document}. This result extends author’s previous works on the long-time behavior of a single point particle. New difficulties arise in the derivation of pointwise estimates of Green’s functions due to infinite reflections of waves in-between the point particles. In particular, the differential equation technique used in previous works alone does not suffice. We overcome this by carefully analyzing the structure of Green’s functions in the Laplace variable, especially their asymptotic and analyticity properties.