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This paper is concerned with a diffuse interface model called asNavier-Stokes/Cahn-Hilliard system. This model is usually used to describe themotion of immiscible two-phase flow with diffusion interface. For the periodicboundary value problem of this system in torus $\mathbb{T}^3$, we prove thatthere exists a global unique strong solution near the phase separation state,which means no vacuum, shock wave, mass concentration, interface collision andrupture will be developed in finite time. Furthermore, we established the largetime behavior of these global strong solution of this system. In particular, wefind that the phase field decays algebraically to the phase separation state.

Asymptotic Stability of Phase Separation States for Compressible Immiscible Two-Phase Flow with Periodic Boundary Condition in 3D