Wellposedness of some Oldroyd models that lack explicit dissipation

We review an argument of Renardy proving existence and regularity for a subset of a class of models of non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwellian models. We propose an effective method for solving these models, including a variational formulation suitable for finite element computation. We consider some model equations proposed for non-Newtonian fluids that are a subset of the Oldroyd models [20]. This includes the upper-convected and lower-convected Maxwellian models. Our objective is to extend the existence proof of Renardy [22] for these equations in various ways. In particular, we show that a variant of his proof can be the basis for an effective solution algorithm. The subset of the Oldroyd models that we study involves three parameters, the fluid kinematic viscosity η and two rheological parameters λ1 and μ1. We will refer to this subset as the “three-parameter” subset. There is another subset of the Oldroyd models [20] for which wellposedness has been established [7]. We will refer to this subset as the “five-parameter” subset as they involve two additional rheological parameters λ2 and μ2. The techniques used for these models are quite different from the ones used by Renardy [22] and are revisited here. The approaches are complementary, and this potentially reflects significant differences in these models. In particular, the techniques in [7] explicitly require λ2 6= 0, and (as far as we are aware) the bounds obtained would degenerate as λ2 → 0. The condition λ2 > 0 leads to an explicit dissipation term that is used in obtaining bounds. When λ2 = 0, such explicit dissipation is missing. For reasons that we will explain, we are forced to limit our approach to the case λ2 = μ2 = 0. Thus there is an interesting question regarding obtaining bounds that hold uniformly for λ2 and μ2 small. 1 Notation We assume that the fluid domain D ⊂ R is connected and has a boundary ∂D with different degrees of regularity for different results. For simplicity, we assume that the boundary conditions