Incompressible Limit of Strong Solutions to the Diffuse Interface Model for Two-phase Flows

This paper is concerned with the incompressible limit problem for strong solutions of compressible two-phase flow models under periodic boundary conditions, where the Navier-Stokes equations are nonlinearly coupled with either Cahn-Hilliard equations or Allen-Cahn equations. The viscosity coefficients are allowed to depend both on the density and the phase field variable. We establish rigorous convergence of both local and global strong solutions of compressible systems to their incompressible systems as the Mach number tends to zero.This theoretical framework establishes an essential linkage between compressible and incompressible phase field models, demonstrating that both formulations exhibit consistent physical fidelity in capturing interfacial flow dynamics.Furthermore, we provide some convergence rate estimates of the solutions.