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

We show that the 2D Euler equations are not locally well-posed in the sense of Hadamard in the \(C^1\) space and in the Besov space \(B^1_{\infty ,1}\). Our approach relies on the technique of Lagrangian deformations of Bourgain and Li (Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces. arXiv:1307.7090). We show that the assumption that the data-to-solution map is continuous in either \(C^1\) or \(B^1_{\infty ,1}\) leads to a contradiction with a result in \(W^{1,p}\) of Kato and Ponce (Rev Mat Iberoam 2:73–88, 1986).