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 R d with any d ≥ 3 is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain-Li [3] in the case d = 3. The ill-posedness is shown for certain critical Lorentz spaces as well.