  We consider the incompressible Euler equation on an analytic domain $\Omega $
with nonhomogeneous boundary condition $u\cdot \mathsf{n} = \overline{u} \cdot
\mathsf{n}$ on $\partial \Omega$, where $\overline{u}$ is a given
divergence-free analytic vector field. We establish local well-posedness for
$u$ in analytic spaces without any compatibility conditions in all space
dimensions. We also prove global well-posedness in the 2D case if
$\overline{u}$ decays in time sufficiently fast.
