The scientific interest in pedestrian crowd motion has grown since the mid 90's, and offers applied mathematicians a variety of challenging problems and interesting applications. As an example, we will discuss the Hughes model, which is one of several PDE models for room evacuation. The model aims to describe pedestrian evacuation in a two-dimensional domain with possibly several exits. Pedestrians are assumed to choose the exit they perceive to be closest, while also avoiding crowded regions. An advantage of the of the model is that pedestrians evaluate in real time their preferred exit and path, which may change over time. The model therefore allows for a complicated and realistic description of pedestrian movement. We will discuss the one-dimensional (in space) variant of the model, which greatly simplifies the original two-dimensional formulation, but which still offers interesting technical challenges. The model takes the form of a scalar conservation law with a discontinuous flux function. I discuss some challenges related to the well-posedness of the one-dimensional formulation.
