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In this paper we mainly study the long time behaviour of solutions to thefinite extensible nonlinear elastic (FENE) dumbbell model with dimension two inthe co-rotation case. Firstly, we obtain the $L^2$ decay rate of the velocityof 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 smallcondition. Our obtained sharp result improves considerably the recent resultsin \cite{Luo-Yin,Schonbek}.

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