On the L^∞(0,T;L^2(Ω)^d)-stability of Discontinuous Galerkin schemes for incompressible flows
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The property that the velocity u belongs to L^∞(0,T;L^2(Ω)^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^∞(0,T;L^2(Ω)^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≥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
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