Euler equation in a channel in space dimension 2 and 3

In this article we consider the Euler equations of an ideal incompressible fluid in a $2D$ and $3D$ channel and we prove the existence and uniqueness of classical solutions for all time for the $2D$ case and the local in time existence for the $3D$ case. For the $2D$ case, the proof makes use of the Schauder fixed point, and specific properties of the Green function in a channel are derived. For the $3D$ case, we use a priori estimates on some appropriate Sobolev spaces and the existence of solution follows by the Galerkin method.