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We study the incompressible Euler equations in the $\alpha$-modulation$M^{s,\alpha}_{p,q}$ and H\"older $C^{1+\sigma}$ spaces on the plane. We showthat for these spaces the associated data-to-solution map is not continuous onbounded sets.

Loss of continuity of the solution map for the Euler equations in $\alpha$-modulation and H\"older spaces