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

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}.