Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system

The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier--Stokes equations, which are coupled with the Cahn--Hilliard equation associated to the Ginzburg--Landau free energy with physically relevant logarithmic potential. This model is studied in bounded smooth domains in $\mathbb{R}^d$, $d=2$, and $d=3$ and is supplemented with a no-slip condition for the velocity, homogeneous Neumann boundary conditions for the order parameter and the chemical potential, and suitable initial conditions. We study uniqueness and regularity of weak and strong solutions. In a two-dimensional domain, we show the uniqueness of weak solutions and the existence and uniqueness of global strong solutions originating from an initial velocity ${\it u}_0 \in {\mathbf{V}}_\sigma$, namely, $\textbf{\textit{u}}_0\in \mathbf{H}_0^1(\Omega)$ such that $\mathrm{div}\, {\it u}_0=0$. In addition, we prove further regularity properties and the validity of the instantaneous separation property. In a three-dimensional domain we show the existence and uniqueness of local strong solutions with initial velocity ${\it u}_0 \in {\mathbf{V}}_\sigma$.