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.

翻译：本文针对完全非齐次的$p$-Navier-Stokes型方程组，提出了一种局部间断Galerkin（LDG）近似方法。基于原始变分形式，我们证明了该方法的适定性、稳定性（先验估计）以及弱收敛性。为此，我们提出了一种新的对流项DG离散格式，并发展了一套抽象的非协调伪单调性理论，并将其应用于所研究的问题。我们还采用该方法处理了$p$-Stokes问题。