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

We study the 3-D incompressible inviscid Oldroyd-B model with fractional stress diffusion $(-\mathrm{\Delta }{)}^{\beta }\tau$ and without stress damping. For $\beta \in (\frac{1}{2},1)$ and $s\in [2\beta +\frac{1}{2},\frac{5}{2}]$, we establish local well-posedness for initial data $(u_0,{\tau }_0)$ in $H^s({\mathbb{R}}^3)$, provided that $\mathrm{\parallel }(u_0,{\tau }_0){\mathrm{\parallel }}_{H^{\beta }}$ is bounded and $\mathrm{\parallel }{\mathrm{\Lambda }}^{\beta }(u_0,{\tau }_0){\mathrm{\parallel }}_{H^{s-\beta }}$ is sufficiently small. If, in addition, $\beta \in (\frac{1}{2},\frac{5}{6}]$, 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 $s\in [2\beta +\frac{1}{2},\frac{5}{2}]$, below the threshold $s>\frac{5}{2}$ considered there. We also provide an ${\mathrm{\ell }}^2$ alternative to the high-frequency ${\mathrm{\ell }}^1$ summability condition in the framework of Wu and Zhao \cite{WuZhao2022}.