Global existence and optimal decay rate to the compressible FENE dumbbell model.

In this paper, we are concerned with global well-posedness and optimal decay rate for strong solutions of the compressible finite extensible nonlinear elastic (FENE) dumbbell model. For d ≥ 2 , we firstly prove that the compressible FENE dumbbell model admits the unique global strong solutions provided initial data are close to equilibrium state. Then, by the Littlewood-Paley decomposition theory and the Fourier splitting method, we show optimal L 2 decay rate of global strong solutions for d ≥ 3. Finally, we mainly study optimal decay rate to the 2-D FENE dumbbell model. The improved Fourier splitting method yields that the L 2 decay rate is ln − l ⁡ (e + t) for any l ∈ N. By virtue of the time weighted energy estimate, we can improve the decay rate to (1 + t) − 1 4 . Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov spaces with negative index and obtain optimal L 2 decay rate without the smallness restriction of low frequencies. [ABSTRACT FROM AUTHOR]