The Compressible Navier-Stokes Equations on the Multi-Connected Domains

Abstract

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.