Optimal decay rate for the generalized Oldroyd-B model with only stress tensor diffusion in ${\mathbb{R}}^2$

Abstract

In this paper, we are concerned with optimal decay rate for the 2-D generalized Oldroyd-B model with only stress tensor diffusion $(-\Delta {)}^{\beta }\tau$. In the case $\beta =1$, we first establish optimal decay rate in $H^1$ framework and remove the smallness assumption of low frequencies by virtue of the Fourier splitting method and the Littlewood-Paley decomposition theory. Furthermore, we prove optimal decay rate for the highest derivative of the solution by a different method combining time frequency decomposition and the time weighted energy estimate. In the case $\frac{1}{2}\le \beta <1$, we study optimal decay rate for the highest derivative of the solution by the improved Fourier splitting method.

24 pages. arXiv admin note: text overlap with arXiv:2110.12330