REFINED POINTWISE ESTIMATES FOR A 1D VISCOUS COMPRESSIBLE FLOW AND THE LONG-TIME BEHAVIOR OF A POINT MASS (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

We present results on the long-time behavior of a point mass moving in a ID viscous compressible fluid. In a previous work, we showed that the velocity V(t) of the point mass decays at least as t⁻³/² . In this note, we give a necessary and sufficient condition on the initial data for the decay rate 3/2 to be optimal. This result is obtained as a corollary to refined pointwise estimates for solutions to the barotropic compressible Navier-Stokes equations. This note is a résumé of the preprint [K. Koike, Refined pointwise estimates for the solutions to the one-dimensional barotropic compressible Navier-Stokes equations: An application to the analysis of the long-time behavior of a moving point mass, https://arxiv.org/abs/2010.06578v1 (2020).] with some numerical results added. Our intention is to explain, in a concise manner, the core idea behind the somewhat lengthy calculations given there.