The velocity $V(t)$ of a point mass moving in a 1D viscous compressible barotropic fluid satisfies a decay estimate $V(t)=O(t^{-3/2})$ [K. Koike, J. Differ. Equ. 271 (2021) 356--413]. In the first part of the talk, I present some numerical simulations suggesting that the estimate $V(t)=O(t^{-3/2})$ is in fact optimal. Then, in the second part, I present mathematical results that give a simple necessary and sufficient condition on the initial data for a lower bound of the form $t^{-3/2}\lesssim |V(t)|$ to hold [K. Koike, arXiv:2010.06578 (2020)]. These results are obtained as corollaries to a theorem on refined pointwise estimates for 1D viscous compressible flow. The key concept of the theorem is the introduction of "bi-diffusion waves" that, together with the well-known diffusion waves, give a precise description of long-time asymptotics of solutions.

Note: This seminar will be held as a Zoom online seminar.