  This paper is devoted to the analysis of an energy-stable discontinuous
Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations within
a decoupled splitting framework. We show that the proposed scheme is uniquely
solvable and mass conservative. The energy dissipation and the $L^\infty$
stability of the order parameter are obtained under a CFL condition. Optimal a
priori error estimates in the broken gradient norm and in the $L^2$ norm are
derived. The stability proofs and error analysis are based on induction
arguments and do not require any regularization of the potential function.
