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

TL;DR

This work analyzes the uniform vanishing damping limit for the 2D inviscid Oldroyd-B model with fractional diffusion in the stress tensor. It combines global regularity results with an enhanced Fourier splitting approach to establish decay rates that are uniform in the damping parameter and to obtain sharp decay estimates for the stress trace. The authors prove global well-posedness for small data in Sobolev spaces, derive optimal decay rates for the damping-free case, and quantify the convergence as the damping parameter vanishes, including precise rates and sharpness results. The findings elucidate how fractional diffusion shapes long-time behavior and provide a rigorous framework for uniform damping vanishing in viscoelastic flows.

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 $\mathrm{tr}τ$ with $a\in (0,1]$, which ensure the sharp uniform damping vanishing rates of $\mathrm{tr}τ$.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• Lemma 2.2