Global Well-Posedness for the 3-D Inviscid Oldroyd-B Model with Fractional Stress Diffusion

Abstract: We study the 3-D incompressible inviscid Oldroyd-B model with fractional stress diffusion ((-Δ)^βτ) and without stress damping. For $β\in (\frac{1}{2},1)$ and $s\in [2β+\frac{1}{2},\frac{5}{2}]$, we establish local well-posedness for initial data $(u_0,τ<em>0)$ in (H^s(\mathbb R^3)), provided that (|(u_0,τ_0)|{H^β}) is bounded and (|Λ^β(u_0,τ0)|{H^{s-β}}) is sufficiently small. If, in addition, $β\in(\frac{1}{2},\frac{5}{6}]$, we give a global well-posedness result. Our results complement the work of Constantin, Wu, Zhao and Zhu \cite{ConstantinWuZhaoZhu2021} by treating the Sobolev regularity $s\in[2β+\frac12,\frac{5}{2}]$, below the threshold $s&gt;\frac52$ considered there. We also provide an $\ell<sup>2$ alternative to the high-frequency $\ell<sup>1$ summability condition in the framework of Wu and Zhao \cite{WuZhao2022}.