Regularity of the solutions for micro–macro models near equilibrium

In this paper, we study the regularity of solutions to two microscopic–macroscopic models: Hookean spring model and FENE dumbbell model. By making use of the structure of Fokker–Planck operator and establishing the general Hardy–Littlewood's inequality, we show that the classical solutions near the equilibrium obtained in [F. Lin, C. Liu, Ping Zhang, On a micro–macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math. 60 (6) (2007) 838–866] and [F. Lin, Ping Zhang, The FENE dumbbell model near equilibrium, Acta Math. Sinica (Chin. Ser.) 24 (2008) 529–538] become smooth with respect to the end-to-end vector variable q for any positive time.