  The inviscid multi-layer quasi-geostrophic equations are considered over an
arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions
are imposed to accommodate the free fluctuations of the top and layer
interfaces. Using the barotropic and baroclinic modes in the vertical
direction, the elliptic system governing the streamfunctions and the potential
vorticity is decomposed into a sequence of scalar elliptic boundary value
problems, where the regularity theories from the two-dimensional case can be
applied. With the initial potential vorticity being essentially bounded, the
multi-layer quasi-equations are then shown to be globally well-posed, and the
initial and boundary conditions are satisfied in the classical sense.
