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

TL;DR

This paper investigates the long-time dynamics of multiple point particles in a 1D viscous compressible fluid, revealing a universal power-law decay of velocities and addressing complex wave interactions.

Contribution

It extends previous single-particle results to multiple particles and develops new analytical techniques for Green's functions in this context.

Findings

Particle velocities decay as t^{-3/2} over time

Analysis of Green's functions reveals asymptotic and analyticity properties

Overcomes difficulties due to wave reflections between particles

Abstract

We study the long-time behavior of \textit{several} point particles in a 1D viscous compressible fluid. It is shown that the velocities of the point particles all obey the power law $t^{-3/2}$. This result extends author's previous works on the long-time behavior of a \textit{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.