On the well-posedness of the incompressible Euler Equation

In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces $H^s({\mathbb{R}}^n)$, $n\ge 2$ and $s>n\mathrm{/}2+1$, can be expressed as a geodesic equation on an infinite dimensional manifold. As an application of this geometric formulation we prove that the solution map of the incompressible Euler equation, associating intial data in $H^s({\mathbb{R}}^n)$ to the corresponding solution at time $t>0$, is nowhere locally uniformly continuous and nowhere differentiable.