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

Abstract

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 $s\in (0,3/2)$ and $\varepsilon >0$, we construct a divergence-free initial vorticity $\omega_0$ defined in $\mathbb{R}^{3}$ satisfying $\| \omega_0 \|_{H^s}\leq \varepsilon$, 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 ) \not \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 $.