Local Well‐Posedness of a Perturbed Problem for the Abels–Garcke–Grün Model in Three Dimensions

ABSTRACT We investigate the Abels–Garcke–Grün model that describes the motion of two viscous incompressible fluids with unmatched densities in the presence of a uniform gravitational field. For the perturbed system with respect to a given equilibrium state in three dimensions, we establish the local existence and uniqueness of a strong solution using a suitable iteration scheme and the energy method. This work lays the foundation for further studies on the Rayleigh–Taylor instability problem of nonhomogeneous two‐phase flows within the framework of diffuse interface models.