Local ill-posedness of the Euler equations in $B_{\infty ,1}^1$

Abstract

We show that the incompressible Euler equations on ${\mathbb{R}}^2$ are not locally well-posed in the sense of Hadamard in the Besov space $B_{\infty ,1}^1$. Our approach relies on the technique of Lagrangian deformations of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in $B_{\infty ,1}^1$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.

arXiv admin note: substantial text overlap with arXiv:1405.1943

References in corpus (3)

• Ill-posedness results in critical spaces for some equations arising in hydrodynamics
• Strong Ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
• Ill-posedness of the incompressible Euler equations in the $C^1$ space

Cited by in corpus (2)

• 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