Equilibration and convected limit in 2D-1D corotational Oldroyd's fluid-structure interaction

TL;DR

This paper analyzes the decay to equilibrium and the limiting behavior of solutions in a coupled 2D-1D Oldroyd fluid-structure interaction system, revealing convergence properties as relaxation time varies.

Contribution

It establishes exponential decay rates to equilibrium and demonstrates convergence to a weak solution in the infinite relaxation time limit for a coupled fluid-structure model.

Findings

Exponential decay rate to equilibrium is independent of initial data.

Solutions converge to a weak solution as relaxation time approaches infinity.

Weak-strong uniqueness holds in the limiting system.

Abstract

We consider a solute-solvent-structure mutually coupled system of equations given by an Oldroyd-type model for a two-dimensional dilute corotational polymer fluid with solute diffusion and damping that is interacting with a one-dimensional viscoelastic shell. Firstly, we give the rate at which its solution decays exponentially in time to the equilibrium solution, independent of the choice of the initial datum. Secondly, as the polymer relaxation time goes to infinity (or, equivalently, the center-of mass diffusion goes to zero), we show that any family of strong solutions of the system described above, that is parametrized by the relaxation time, converges to an essentially bounded weak solution of a corotational polymer fluid-structure interaction system whose solute evolves according to the convected time derivative of its extra stress tensor. A consequence of this is a weak-strong uniqueness result.