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

TL;DR

This paper refines pointwise estimates for solutions to 1D barotropic compressible Navier-Stokes equations, providing a necessary and sufficient condition for the optimal decay rate of a moving point mass's velocity over time.

Contribution

It introduces refined estimates using inter-diffusion and diffusion waves, improving understanding of the long-time behavior of a point mass in a viscous compressible fluid.

Findings

Decay estimate V(t)=O(t^{-3/2}) is optimal under certain initial data conditions.

Refined pointwise estimates improve the approximation of fluid behavior around the point mass.

Necessary and sufficient conditions for the decay rate are established.

Abstract

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.