Large time behavior of global strong solutions to the 2-D compressible FENE dumbbell model

TL;DR

This paper investigates the long-term decay behavior of strong solutions to the 2-D compressible FENE dumbbell model, establishing decay rates using Fourier splitting, energy estimates, and Littlewood-Paley theory.

Contribution

It provides new decay rate results for solutions, including optimal decay in Besov spaces, for the 2-D compressible FENE dumbbell model.

Findings

$L^2$ decay rate is $ ext{ln}^{-l}(e+t)$ for any $l eq 0$

Improved decay rate to $(1+t)^{-rac{1}{4}}$ using energy estimates

Solutions belong to Besov spaces with negative index, achieving optimal decay

Abstract

In this paper, we mainly study large time behavior of the strong solutions to the 2-D compressible finite extensible nonlinear elastic (FENE) dumbbell model. The Fourier splitting method yields that the $L^2$ decay rate is $\ln^{-l}(e+t)$ for any $l\in\mathbb{N}$. By virtue of the time weighted energy estimate, we can improve the decay rate to $(1+t)^{-\frac{1}{4}}$. Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov space with negative index and obtain the optimal $L^2$ decay rate.