Long-time behavior for the two-dimensional motion of a disk in a viscous fluid

TL;DR

This paper investigates the long-term behavior of a disk moving in a viscous fluid in two dimensions, deriving decay estimates for the fluid-structure system modeled by Navier-Stokes equations.

Contribution

It provides new $L^p$-$L^q$ decay estimates and asymptotic expansions for the solutions of the two-dimensional fluid-rigid disk problem.

Findings

Derived $L^p$-$L^q$ decay estimates for linearized equations

Computed the first term in the asymptotic expansion of solutions

Established time-decay estimates for the full nonlinear system

Abstract

In this article, we study the long-time behavior of solutions of the two-dimensional fluid-rigid disk problem. The motion of the fluid is modeled by the two-dimensional Navier-Stokes equations, and the disk moves under the influence of the forces exerted by the viscous fluid. We first derive $L^p$-$L^q$ decay estimates for the linearized equations and compute the first term in the asymptotic expansion of the solutions of the linearized equations. We then apply these computations to derive time-decay estimates for the solutions to the full Navier-Stokes fluid-rigid disk system.