Contribution to nonlinear and nonlocal partial differential equations and application to road traffic.

Summary This thesis deals with the modeling, analysis and numerical analysis of nonlinear and nonlocal partial differential equations with applications to road traffic. Road traffic can be modeled at different scales. In particular, one can consider the microscopic scale which describes the dynamics of each vehicle individually and the macroscopic scale which sees traffic as a fluid and describes traffic using macroscopic quantities such as vehicle density and average speed. In this thesis, using viscosity solution theory, we make the transition from microscopic to macroscopic models. The interest of this switch is that microscopic models are more intuitive and easy to handle to simulate particular situations (junctions, traffic lights,.) but they are not adapted to large simulations (to simulate the traffic in a whole city for example). On the contrary, macroscopic models are less easy to modify (to simulate a particular situation) but they can be used for large scale simulations. The idea is therefore to find the macroscopic model equivalent to a microscopic model that describes a specific scenario (a junction, a fork, different types of drivers, a school zone,.). The first part of this thesis contains a homogenization and numerical homogenization result for a microscopic model with different types of drivers. In a second part, homogenization and numerical homogenization results are obtained for microscopic models with a local disturbance (speed bump, school zone,.). Finally, we present a homogenization result in the context of a bifurcation.