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

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.

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