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

TL;DR

This paper demonstrates that the incompressible Euler equations are ill-posed in the Besov space $B^1_{ abla,1}$ on $ abla^2$, showing that the data-to-solution map cannot be continuous in this space, contrasting with known well-posedness in $W^{1,p}$.

Contribution

It proves local ill-posedness of the Euler equations in the Besov space $B^1_{ abla,1}$ using Lagrangian deformation techniques, extending understanding of regularity thresholds.

Findings

Euler equations are not locally well-posed in $B^1_{ abla,1}$.

Continuity of the data-to-solution map in $B^1_{ abla,1}$ leads to contradiction.

Well-posedness in $W^{1,p}$ does not extend to $B^1_{ abla,1}$.

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 $B^1_{\infty,1}$. 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 $B^1_{\infty,1}$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.