  In the present paper, we propose a Local Discontinuous Galerkin (LDG)
approximation for fully non-homogeneous systems of $p$-Navier-Stokes type. On
the basis of the primal formulation, we prove well-posedness, stability (a
priori estimates), and weak convergence of the method. To this end, we propose
a new DG discretization of the convective term and develop an abstract
non-conforming theory of pseudo-monotonicity, which is applied to our problem.
We also use our approach to treat the $p$-Stokes problem.
