Ill-posedness of the incompressible Euler equations in the $C^1$ space

Abstract

We prove that the 2D Euler equations are not locally well-posed in $C^1$. Our approach relies on the technique of Lagrangian deformations and norm inflation of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in $C^1$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.

Cited by in corpus (4)

• Ill-posedness results in critical spaces for some equations arising in hydrodynamics
• Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces
• Loss of continuity of the solution map for the Euler equations in $\alpha$-modulation and Hölder spaces
• Local ill-posedness of the Euler equations in $B_{\infty ,1}^1$