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

Abstract

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in ${\mathbb{R}}^d$ with any $d\ge 3$ 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.

To appear in J. Funct. Anal