The Liouville theorem and the $L^2$ decay of the FENE dumbbell model of polymeric flows

In this paper we mainly investigate the finite extensible nonlinear elastic (FENE) dumbbell model with dimension $d\ge 2$ in the whole space. We first proved that there is only the trivial solution for the steady-state FENE model under some integrable condition. Our obtained results generalize and cover the classical results to the stationary Navier-Stokes equations. Then, we study about the $L^2$ decay of the co-rotation FENE model. Concretely, the $L^2$ decay rate of the velocity is $(1+t{)}^{-\frac{d}{4}}$ when $d\ge 3$, and ${\ln }^{-k}(e+t),k\in \text{mathds}{N}^{+}$ when $d=2$. This result improves considerably the recent result of \cite{Schonbek2} by Schonbek. Moreover, the decay of general FENE model has been considered.