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

Abstract

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 show that for these spaces the associated data-to-solution map is not continuous on bounded sets.