The inviscid inflow-outflow problem via analyticity

We consider the incompressible Euler equation on an analytic domain $\mathrm{\Omega }$ with nonhomogeneous boundary condition $u\cdot \mathsf{n}=\bar{u}\cdot \mathsf{n}$ on $\mathrm{\partial }\mathrm{\Omega }$, where $\bar{u}$ 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 $\bar{u}$ decays in time sufficiently fast.