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

TL;DR

This paper investigates the optimal decay rates for solutions to the 2-D generalized Oldroyd-B model with stress tensor diffusion, employing advanced mathematical techniques to improve understanding of the model's long-term behavior.

Contribution

It establishes optimal decay rates in different diffusion regimes and removes smallness assumptions using Fourier analysis and energy estimates.

Findings

Optimal decay rate in $H^1$ for $eta=1$ without smallness assumption

Decay rates for highest derivatives using time frequency decomposition

Improved decay analysis for $1/2 \,\leq\, \beta < 1$

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 $\frac 1 2\leq \beta <1$, we study optimal decay rate for the highest derivative of the solution by the improved Fourier splitting method.