  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.
