Global Classical Solutions to the Viscous Two-Phase Flow Model with Navier-type Slip Boundary Condition in 2D Bounded Domains

We consider the viscous two-phase flow model with Navier-type slip boundary condition in a two-dimensional simply connected bounded domain with C^∞ boundary ∂Ω . Based on some new estimates of effective viscous flux on boundary integrals related to the Navier-type slip boundary condition, we establish the global existence and large time behavior of the classical solutions to two-phase flow model in time provided the initial energy is suitably small even if the density contains vacuum and has large oscillations. This is the first result concerning the global existence of classical solutions to the viscous two-phase flow model with density containing vacuum initially for general 2D bounded smooth domain.