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

TL;DR

This paper demonstrates that the data-to-solution map for the incompressible Euler equations is discontinuous in certain $ ext{α}$-modulation and H"older spaces, highlighting limitations in solution stability within these function spaces.

Contribution

It establishes the discontinuity of the solution map for Euler equations in specific modulation and H"older spaces, revealing new insights into solution behavior in these function spaces.

Findings

Discontinuity of the data-to-solution map in $ ext{α}$-modulation spaces.

Discontinuity of the data-to-solution map in H"older spaces.

Implications for stability analysis of Euler solutions.

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.