Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

Abstract

We consider the incompressible Euler equations on RdRd or TdTd, where d∈{2,3}d∈{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 a1a1 and Sobolev regularity in the labels a2,…,ada2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.