Asymptotic analysis in flow curves for a model of soft glassy fluids

In this article, we rigourously prove several asymptotical results for the flow curves of the Hebraud-Lequeux model, a rheological model which describes the behaviour of soft glassy fluids. This model has a control parameter alpha which governs the behaviour of the fluid at low shear rate. More precisely, we consider tau((gamma) over dot) the stress in a block that is sheared at a constant rate (gamma) over dot and we prove that the system exhibits a transition in its behaviour at low shear rate when a goes through a critical value. The study is complicated by the fact that one of the parameter is only given implicitly and also we have to study two variable function in the neighbourhood of singularities.