Local well-posedness of a perturbation problem for the Abels-Garcke-Gr\"un model in three dimensions

TL;DR

This paper proves local existence and uniqueness of strong solutions for a three-dimensional two-fluid model with gravity, providing a basis for future instability analysis in nonhomogeneous flows.

Contribution

It establishes the local well-posedness of a perturbation problem for the Abels-Garcke-Grün model in three dimensions, a novel result in the study of two-phase flow models.

Findings

Proved local existence and uniqueness of strong solutions.

Developed an iteration scheme and energy method for the analysis.

Set the stage for future Rayleigh-Taylor instability studies.

Abstract

We investigate the Abels-Garcke-Gr\"un model that describes the motion of two viscous incompressible fluids with unmatched densities in the presence of a uniform gravitational field. For the perturbated 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.