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

A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part I: Convergence analysis