  We consider the incompressible 2D Euler equations on bounded spatial domain
$S$, and study the solution map on the Sobolev spaces $H^k(S)$ ($k > 2$).
Through an elaborate geometric construction, we show that for any $T >0$, the
time $T$ solution map $u_0 \mapsto u(T)$ is nowhere locally uniformly
continuous and nowhere Fr\'echet differentiable.
