Loss of continuity of the solution map for the Euler equations in $α$-modulation and Hölder spaces

Abstract

We study the incompressible Euler equations in the $α$-modulation $M^{s,α}_{p,q}$ and Hölder $C^{1+σ}$ spaces on the plane. We show that for these spaces the associated data-to-solution map is not continuous on bounded sets.