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

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.