Existence and Stability of Strong Solutions to the Abels-Garcke-Grün model in Three Dimensions

Abstract

This work is devoted to the analysis of 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 $({\mathbf{u}}_0,{\varphi }_0)\in {\mathbf{H}}_{\sigma }^1\times H^2(\Omega )$ 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 {\it Calc. Var. (2021) 60:100} to three dimensional bounded domains.