Galilean Boost and Non-uniform Continuity for Incompressible Euler

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(ℝ^d) , s ≥ 0 . This settles the end-point case ( s = 0) left open in Himonas–Misiołek (Commun Math Phys 296(1):285–301, 2010 ) and gives a unified treatment for all H s . We also show the solution map is nowhere uniformly continuous.