Uniform vanishing damping limit for the 2D inviscid Oldroyd-B model with fractional stress tensor diffusion

TL;DR

This paper investigates the uniform vanishing damping limit of the 2D inviscid Oldroyd-B model with fractional stress tensor diffusion, establishing optimal and uniform decay rates and improving spectral analysis results.

Contribution

It introduces fractional stress tensor diffusion to the 2D Oldroyd-B model and proves uniform decay rates as damping vanishes, enhancing understanding of the model's long-term behavior.

Findings

Fractional diffusion reduces global regularity requirements.

Established optimal decay rates for zero damping coefficient.

Proved uniform decay rates as damping parameter approaches zero.

Abstract

This paper is devoted to the uniform vanishing damping limit of the 2D inviscid Oldroyd-B model with fractional stress tensor diffusion. Firstly, we find that fractional stress tensor diffusion helps to reduce the global regularity of the 2D Oldroyd-B model with damping coefficient $a\in[0,1]$. By virtue of improved Fourier splitting method, we then prove the optimal time decay rates under the critical regularity for $a=0$. When $a\in (0,1]$, we establish time decay rates that are uniform with respect to $a$. Combining the time decay rate for $a\in [0,1]$ and the time integrability, we obtain the uniform damping vanishing rates for the 2D Oldroyd-B model. Using spectral analysis methods, we finally improve the time decay rates for $\mathrm{tr}\tau$ with $a\in (0,1]$, which ensure the sharp uniform damping vanishing rates of $\mathrm{tr}\tau$.