Existence and Stability of Strong Solutions to the Abels-Garcke-Gr\"{u}n model in Three Dimensions

TL;DR

This paper proves the existence, local uniqueness, and stability of strong solutions to the three-dimensional Abels-Garcke-Grün (AGG) model, extending previous two-dimensional results to more realistic 3D bounded domains.

Contribution

It establishes the existence, local uniqueness for separated initial data, and stability estimates for strong solutions of the 3D AGG model, advancing mathematical understanding of this complex fluid system.

Findings

Existence of local-in-time strong solutions from initial data.

Local uniqueness for initial data separated from pure phases.

Stability estimates between AGG and model H solutions.

Abstract

This work is devoted to the analysis of strong solutions to the Abels-Garcke-Gr\"{u}n (AGG) model in three dimensions. First, we prove the existence of local-in-time strong solutions originating from an initial datum $(\mathbf{u}_0, \phi_0)\in \mathbf{H}^1_\sigma \times H^2(\Omega)$ such that $\mu_0 \in H^1(\Omega)$ and $|\overline{\phi_0}|\leq 1$. For the subclass of initial data that are strictly separated from the pure phases, the corresponding strong solutions are locally unique. Finally, we show a stability estimate between the solutions to the AGG model and the model H. These results extend the analysis achieved by the author in {\it Calc. Var. (2021) 60:100} to three dimensional bounded domains.