Principal investigator: Verena Bögelein
Staff member: Michael Strunk

The project will be funded by the FWF (Fonds zur Förderung der wissenschaftlichen Forschung). 

Description of the project:

The partial differential equations considered model problems of optimal transportation with congestion effects. The model is based on a game-theoretical approach to traffic dynamics, the so-called Wardrop equilibrium. The wardrop equilibrium is based on two principles: User equilibrium, which assumes that each user chooses the best possible route, and system optimality, which assumes that users behave cooperatively so that the average travel time is minimal.

Instead of studying the effects on the real traffic dynamics, in this project we are interested in the associated partial differential equation and its solutions. In particular, we will investigate regularity properties of the solutions. Our aim is to systematically investigate higher regularity properties, i.e. regularity beyond Lipschitz continuity. We will consider interior and boundary regularity, the scalar and the vectorial case as well as optimality aspects. The methods used to solve these problems are manifold. Deep knowledge of real analysis and regularity theory for nonlinear PDEs is required.

The class of partial differential equations under consideration is called strongly degenerate PDEs. There is also a time-dependent, parabolic counterpart. This parabolic PDE appears in models of gas filtration with nonlinear effects, where the flow only starts at a certain critical pressure.

There are many important examples of PDEs with degenerate structure, such as the elliptic and parabolic p-Laplace equation, the porous media equation, the Stefan problem, PDEs with vanishing coefficients, etc. Each of these equations has its own peculiarities. Deep analytical techniques are required to understand them. In the last decades, a certain understanding of regularity has been developed for these equations. In contrast, regularity theory for strongly degenerate PDEs is a largely open field. In this project, we systematically investigate the topic to provide a better understanding of PDEs with general degenerate structure.