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In this paper, we mainly study global well-posedness and optimal decay ratefor the strong solutions of the compressible co-rotation finite extensiblenonlinear elastic (FENE) dumbbell model. This model is a coupling of theisentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planckequation. We first prove that the FENE dumbbell model admits a unique globalstrong solution provided the initial data are close to equilibrium state for$d\geq 2$. Moreover, for $d\geq3$, we show that optimal decay rates of globalstrong solutions by the linear spectral theory and a more precise Hardy typeinequality.

Global strong solutions and large time behavior to the compressible co-rotation FENE dumbbell model of polymeric flows near equilibrium