Existence and stability of strong solutions to the Abels–Garcke–Grün model in three dimensions

Abstract

This work is devoted to the analysis of the strong solutions to the Abels–Garcke–Grün (AGG) model in three dimensions. First, we prove the existence of local-in-time strong solutions originating from an initial datum $(u_0,{\varphi }_0)\in {\mathbf{H}}_{\sigma }^1\times H^2(\Omega )$ such that ${\mu }_0\in H^1(\Omega )$ and $|\overline{{\varphi }_0}|\le 1$. 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 2021 from two-dimensional bounded domains to three-dimensional ones.