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

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