The linearized 3d Euler equations with inflow, outflow

In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence uniqueness solutions to 3D Euler equations in which on certain inflow boundary components fluid is forced into domain while other outflow drawn out domain. A key tool they used was linearized vorticity form. We extend their result problem multiply connected domains establish compatibility conditions initial data that allow higher regularity solutions.