Abstract
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).
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Acknowledgments
We thank the anonymous referees for their suggestions in helping to improve the exposition of the paper. GM gratefully acknowledges support from the Simons Center for Geometry and Physics where part of the research for this paper was done. TY was partially supported by JSPS KAKENHI Grant Number 25870004.
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Misiołek, G., Yoneda, T. Local ill-posedness of the incompressible Euler equations in \(C^1\) and \(B^1_{\infty ,1}\) . Math. Ann. 364, 243–268 (2016). https://doi.org/10.1007/s00208-015-1213-0
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DOI: https://doi.org/10.1007/s00208-015-1213-0