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

TL;DR

This paper demonstrates that solutions to the 3D incompressible Euler equations can instantaneously lose Sobolev regularity, showing a sharp regularity breakdown in finite time despite smooth initial conditions.

Contribution

It constructs explicit solutions exhibiting instantaneous Sobolev regularity loss, revealing new insights into the regularity dynamics of the Euler equations.

Findings

Solutions lose Sobolev regularity instantly at t=0

Constructed initial vorticity with small H^s norm

Established uniqueness among certain solution classes

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 $.