Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

TL;DR

This paper compares the regularity properties of incompressible Euler equations in Lagrangian and Eulerian coordinates, showing well-posedness in Lagrangian but not in Eulerian frameworks, with implications for understanding fluid dynamics.

Contribution

It demonstrates that Euler equations are well-posed in Lagrangian coordinates with fixed regularity, but not in Eulerian coordinates, highlighting fundamental differences in coordinate frameworks.

Findings

Lagrangian coordinates yield local well-posedness with fixed analyticity radius.

Lagrangian coordinates allow well-posedness in anisotropic Gevrey and Sobolev spaces.

Eulerian coordinates do not support these regularity results, indicating a fundamental difference.

Abstract

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 $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.