  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}.

Analysis of PDEs