A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces

TL;DR

This paper presents a straightforward proof demonstrating the ill-posedness of the incompressible Euler equations in critical Sobolev spaces for dimensions three and higher, extending previous results to Lorentz spaces.

Contribution

It offers a simplified proof of ill-posedness for Euler equations in critical Sobolev and Lorentz spaces across dimensions d ≥ 3, expanding prior work.

Findings

Ill-posedness established for Euler equations in critical Sobolev spaces

Extension of ill-posedness results to Lorentz spaces

Simplification of the proof method compared to previous approaches

Abstract

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in $\mathbb{R}^{d}$ with any $d\ge3$ is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li in the case $d = 3$. The ill-posedness is shown for certain critical Lorentz spaces as well.