Galilean Boost and Non-uniform Continuity for Incompressible Euler

Abstract

By using an idea of localized Galilean boost, we show that the data-to-solution map for incompressible Euler equations is not uniformly continuous in $${H^s({\mathbb{R}}^d)}$$, $${s \ge 0}$$. This settles the end-point case (s = 0) left open in Himonas–Misiolek (Commun Math Phys 296(1):285–301, 2010) and gives a unified treatment for all Hs. We also show the solution map is nowhere uniformly continuous.