Well-posedness of the two-dimensional Abels-Garcke-Gr\"un model for two-phase flows with unmatched densities

TL;DR

This paper investigates the well-posedness of the two-dimensional Abels-Garcke-Grün model for two-phase flows with different densities, establishing local and global existence, uniqueness, and continuous dependence of strong solutions.

Contribution

It proves the existence, uniqueness, and continuous dependence of strong solutions for the 2D AGG model, including global solutions in periodic domains, advancing understanding of this complex fluid system.

Findings

Local strong solutions exist in bounded domains.

Global strong solutions exist in periodic settings.

Strong solutions are unique and depend continuously on initial data.

Abstract

We study the Abels-Garcke-Gr\"un (AGG) model for a mixture of two viscous incompressible fluids with different densities. The AGG model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a (non-constant) concentration-dependent density and an additional flux term due to interface diffusion. In this paper we address the well-posedness problem in the two-dimensional case. We first prove the existence of local strong solutions in general bounded domains. In the space periodic setting we show that the strong solutions exist globally in time. In both cases we prove the uniqueness and the continuous dependence on the initial data of the strong solutions.