The $L^2$ decay for the 2D co-rotation FENE dumbbell model of polymeric flows

TL;DR

This paper investigates the long-term decay behavior of solutions to the 2D co-rotation FENE dumbbell model, establishing decay rates for velocity and removing small data restrictions using advanced harmonic analysis techniques.

Contribution

It provides the first sharp $L^2$ decay rate for the 2D co-rotation FENE model and removes the small data condition through Littlewood-Paley theory, improving previous results.

Findings

$L^2$ decay rate of velocity is $(1+t)^{-rac{1}{2}}$ for small data

Decay rate is established without small data restriction using Littlewood-Paley theory

Results improve upon previous decay estimates in the literature

Abstract

In this paper we mainly study the long time behaviour of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the $L^2$ decay rate of the velocity of the 2D co-rotation FENE model is $(1+t)^{-\frac{1}{2}}$ with small data. Then, by virtue of the Littlewood-Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in \cite{Luo-Yin,Schonbek}.