INTRODUCTION Hysteresis is a complex physical phenomenon occurring in continuum mechanics, ferromagnetism and filtration through porous media; in particular, it often takes place in phase transitions. The mathematical literature on this subject is very wide and applies to several phenomena by exploiting different models and techniques; we quote for instance . This paper focuses on a very simple model arising in oil recovery that was proposed in . Such a model does not certainly aim at describing accurately the complex dynamic of the problem but rather to highlight some mathematical issues that characterize it. More precisely, we consider a fluid flow in a porous medium, which is constituted by a aqueous phase, here formed by water, and a liquid phase, formed by oil. The flow is modeled by the diffusive equation s_t + f_x=\epsilon s_{xx}, where s is the water saturation, f the water fractional flow and ϵ the (constant) capillarity-induced diffusion coefficient. Both s and f are valued in [0, 1]. Hysteresis comes into the play through the flow f, which does not only depend on s but also on its history and current trend. Roughy speaking, f can be thought as a multi-valued function; for s fixed, these multiple values are parametrized by a new variable π, which encodes the behavior of s in the past and the actual increasing or decreasing of s. As a consequence, an equation for π is introduced. A more precise description of the model is provided in the following section. In the case ϵ = 0, the Riemann problem for equation was briefly studied in ; a sketch of the construction is reported in . The solution to such problem is far from being unique, not only because many combination of waves are possible for the same initial data, but in particular because _hysteresis loops_ appear. However, for constant ϵ > 0, a relaxation approximation for the equation of π is introduced in and the authors determine which shock waves have a diffusive-relaxation profile. The drawback of this construction is that the flux function needs to be extended outside its natural domain, as well as the solutions. Such an approximation violates the famous subcharacteristic condition ; indeed, the failure of this condition is balanced by the presence of the diffusion term, and the whole effect is to allow the hysteresis loops. We mention that more complicated flows have been considered in this framework. For instance, in a three-component, two-phase flow is proposed, where polymer is added to the aqueous phase to increase the viscosity of this phase; in this way the extraction of oil from the porous medium is enhanced. To take into account the presence of gas, which is common in porous rocks, in the authors introduce a three-component, three-phase flow. The plan of the paper now follows[1]. In Section [sec:model] we give more details on the system we introduced quickly above; in particular, we formulate there the assumptions on the flux functions. Weak solutions are presented in Section [sec:Preliminaries], while in Section [sec:basic_waves] we describe the waves admitting viscous profiles. Section [sec:Solutions] we briefly recall the solution to the Riemann problem and illustrate it with several pictures. In the final Section [sec:VV] we discuss the uniqueness of solutions to the Riemann problem. [1] Check at the end