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 $\beta \in (\frac{1}{2},1)$ 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 \(\|(u_0,τ_0)\|_{H^β}\) is bounded and \(\|Λ^β(u_0,τ_0)\|_{H^{s-β}}\) 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 ${\ell }^2$ alternative to the high-frequency ${\ell }^1$ summability condition in the framework of Wu and Zhao \cite{WuZhao2022}.