Non-Uniform Dependence on Initial Data of Solutions to the Euler Equations of Hydrodynamics

We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H ^s (Ω) topology for any ${s \in \mathbb{R}}We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal. More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H ^s (Ω) topology for any s ? \mathbbR{s \in \mathbb{R}} if the domain Ω is the (flat) torus \mathbbTⁿ=\mathbbRⁿ/2p\mathbbZⁿ{\mathbb{T}^n=\mathbb{R}^n/2\pi\mathbb{Z}^n} and for any s > 0 if the domain is the whole space \mathbbRⁿ{\mathbb{R}^n}.