Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

We consider the incompressible Euler equations on ${\mathbb{R}}^d$, 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 well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_d$. © In Eulerian coordinates both results (a) and (b) above are false.