The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(\Omega)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(\Omega)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest.

翻译：速度$\boldsymbol{u}$属于$L^\infty(0,T;L^2(\Omega)^d)$这一性质，是不可压缩流体模型能量解定义中的基本要求。因此，离散化方法产生的解在$L^\infty(0,T;L^2(\Omega)^d)$范数下具有一致稳定性是至关重要的。本文证明，对于具有$p$结构的非牛顿模型，在假设$p\geq \frac{3d+2}{d+2}$的条件下，其时间与空间上的间断伽辽金(Discontinuous Galerkin, DG)离散格式确实满足此稳定性要求；其中时间离散等价于RadauIIA隐式龙格-库塔(Runge-Kutta)方法。我们还证明了数值格式弱收敛于系统的弱解；这种基于求积法的格式收敛性结果似乎具有新颖性。作为辅助结果，我们还在DG空间上推导了Gagliardo-Nirenberg型不等式，该结果可能具有独立的研究价值。