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

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 $\mathrm{tr}\tau$.