Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Abstract

We consider the incompressible Euler equations on ${\mathbb{R}}^d$ or ${\mathbb{T}}^d$, where $d\in \{2,3\}$. We prove that:

(a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius).

(b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces, e.g. Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,\dots ,a_d$.

(c) In Eulerian coordinates both results (a) and (b) above are false.