On the $L^\infty(0,T;L^2(Ω)^d)$-stability of Discontinuous Galerkin schemes for incompressible flows

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 implicitly constituted models with $p$-structure, in general, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to a RadauIIA Implicit Runge-Kutta method. To aid in the proof, we derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest