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

• Published in: Journal of Differential Equations 2024-09, Vol.404, p.130-181
• Main Authors: Luo, Zhaonan, Luo, Wei, Yin, Zhaoyang
• Format: Article
• Language: English
• Subjects: Global existence; The compressible FENE dumbbell model; Time decay rate
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2024.05.044

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 L2 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 L2 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)−14. 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 L2 decay rate without the smallness restriction of low frequencies.