  In this paper, we derive a theoretical analysis of an interior penalty
discontinuous Galerkin methods for solving the Cahn-Hilliard-Navier-Stokes
model problem. We prove unconditional unique solvability of the discrete
system, obtain unconditional discrete energy dissipation law, and derive
stability bounds with a generalized chemical energy density. Convergence of the
method is obtained by proving optimal a priori error estimates. Our analysis of
the unique solvability is valid for both symmetric and non-symmetric versions
of the discontinuous Galerkin formulation.
