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，是定义不可压缩流体模型能量解的基本要求。因此，离散方法所得解在 $L^\infty(0,T;L^2(\Omega)^d)$-范数下的一致稳定性具有重要价值。本文证明，对于具有 $p$ 结构的非牛顿模型（其中 $p\geq \frac{3d+2}{d+2}$），时间与空间上的间断伽辽金（DG）离散格式确实满足这一性质；其时间离散等价于RadauIIA隐式龙格-库塔方法。我们同时证明了数值格式弱收敛于系统的弱解——此类基于求积格式的收敛结果尚属首次。作为辅助结果，我们还推导了DG空间上的Gagliardo-Nirenberg型不等式，该结果可能具有独立的研究价值。