# Refined Pointwise Estimates for 1D Viscous Compressible Flow and the Long-Time Behavior af a Point Particle

## Speaker

Kai Koike, Kyoto University

## Time

2021.01.25 10:00-11:00

Online--ZOOM APP

## ZOOM Info

Conference ID: 936-906-02086
We consider the long-time behavior of a point particle moving in a 1D viscous barotropic fluid. In a previous work, I showed that the velocity of the point particle $V(t)$ satisfies a decay estimate $V(t)=O(t^{-3/2})$. This was shown as a corollary to a theorem on pointwise estimates of solutions to the fluid equations. In this talk, we tackle the following question: Is the rate $-3/2$ optimal or not? To this end, I present a refined version of the pointwise estimates in the previous work, which enables us to derive a necessary and sufficient condition for a lower bound $C^{-1}(t+1)^{-3/2}\leq|V(t)|$ ($t\gg 1, C>1$) to hold. We treat certain waves that emerge from interaction of waves of different speeds as leading order terms, whereas in the previous work, such waves are analyzed but considered as remainder terms of the diffusion wave approximation. (This talk is based on the preprint: https://arxiv.org/abs/2010.06578.