The Optimal \({{\varvec{L}}^2}\) Decay Rate of the Velocity for the General FENE Dumbbell Model

In this paper we mainly study large time behavior for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. The sharp \(L^2\) $L^2$ decay rate was obtained on the co-rotational case. We prove that the optimal \(L^2\) $L^2$ decay rate of the velocity of the general FENE dumbbell model is \((1+t)^{-\frac{d}{4}}\) ${(1+t)}^{-\frac{d}{4}}$ with \(d\ge 2\) $d\ge 2$ . Our obtained result is sharp and improves considerably the previous result in Luo and Yin (Arch Ration Mech Anal 224(1):209–231, 2017).