On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra
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We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator.