  This paper investigates the isentropic compressible Navier-Stokes equations
on k-connected domains under Navier-slip boundary conditions. We study the
multi-solvability of the stationary systems on general domains, which is
closely related with the Cauchy-Riemann systems and critical points of harmonic
functions on the domain. Then based on the structure of Green's functions, the
commutator estimates are obtained on the circular domains and extended to
general domains with the help of conformal mappings. Moreover, we will utilize
these assertions to discuss the global well-posedness and large time behaviours
of the non-stationary systems on general domains with large initial values
containing vacuum.
