标题： 三维不可压缩欧拉方程的瞬时Sobolev正则性丧失 显示英文标题

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

摘要： 我们证明了在$\mathbb{R}^{3}$中，三维不可压缩欧拉方程的超临界 Sobolev 正则性在瞬时和连续时间上都会损失。 即，对于任何$s\in (0,3/2)$和$\varepsilon >0$，我们构造一个在$\mathbb{R}^{3}$中定义的无散初始涡量$\omega_0$，满足$\| \omega_0 \|_{H^s}\leq \varepsilon$，以及$T>0$、$c>0$并且对应一个局部时间解$\omega$，使得对于每个$t\in [0,T]$，$\omega (\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$和$ \omega (\cdot ,t ) \not \in {H^\beta }$对于任何$\beta > \frac{s-ct}{1+ct} $。 此外，$\omega$在所有具有初始条件$\omega_0$的解中是唯一的，这些解在局部$C^2$并且属于$C([0,T];L^p )$对于任何$p>3 $。 显示英文摘要

摘要： 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 $.