Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in ${\mathbb{R}}^3$. Namely, for any $s\in (0,3\mathrm{/}2)$ and $\epsilon >0$, we construct a divergence-free initial vorticity ${\omega }_0$ defined in ${\mathbb{R}}^3$ satisfying $\mathrm{\parallel }{\omega }_0{\mathrm{\parallel }}_{H^s}\le \epsilon$, as well as $T>0$, $c>0$ and a corresponding local-in-time solution $\omega$ such that, for each $t\in [0,T]$, $\omega (\cdot ,t)\in H^{\frac{s-ct}{1+ct}}$ and $\omega (\cdot ,t)\in ̸H^{\beta }$ for any $\beta >\frac{s-ct}{1+ct}$. Moreover, $\omega$ is unique among all solutions with initial condition ${\omega }_0$ which are locally $C^2$ and belong to $C([0,T];L^p)$ for any $p>3$.