  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 domain in 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 u_0
in V, namely u_0 in H_0^1 such that div 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 u_0 in V.
