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

TL;DR

This paper proves the global stability and asymptotic behavior of phase separation states in a 3D compressible two-phase flow model, ensuring no finite-time singularities occur and describing the decay to equilibrium.

Contribution

It establishes the existence, uniqueness, and long-term decay of strong solutions for the Navier-Stokes/Cahn-Hilliard system in 3D with periodic boundary conditions, near phase separation states.

Findings

Existence of global unique strong solutions near phase separation.

Solutions decay algebraically to equilibrium over time.

No finite-time development of singularities like vacuum or interface collision.

Abstract

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