  This paper would focus on the subject of the 2-D incompressible
Navier-Stokes-Cahn-Hilliard (NS-CH) system with a singular free energy density.
Due to lack of the maximum principle for the convective Cahn-Hilliard equation
(as a fourth-order parabolic equation), we construct its approximate
second-order parabolic equation, and use comparison principle and the basic
energy estimates to separate the solution from the singular values of the
singular free energy density, where the Orlicz embedding theorem plays a key
role. Based on these, we prove the global well-posedness of the Cauchy problem
of the 2-D NS-CH equations with periodic domains by using energy estimates and
the Logarithmic Sobolev inequality.
