Global existence of smooth solution to a non-conservative compressible two-fluid model in exterior domain

In this paper, we consider the initial boundary value problem for a non-conservative compressible two-fluid model with Navier-slip boundary conditions in an exterior domain of $R^3$. We prove the global existence of smooth solution in a setting that the capillary pressure $f\left({\alpha }^{-}{\rho }^{-}\right)=P^{+}-P^{-}\ne 0$, where f is assumed to be a strictly decreasing function near the equilibrium of ${\alpha }^{-}{\rho }^{-}$, and prove that this assumption has a critical stabilization effect in the question. Furthermore, we establish the explicit decay rates of smooth solution to its equilibrium state.