Singularity Formation in Nonlinear Wave Equations

Partial differential equations (PDEs) are a fascinating branch of mathematics with many applications in natural sciences, engineering and economics. A special class of PDEs are so-called nonlinear wave equations which are generalizations of the well-known wave equation. Such systems occur frequently in theoretical physics. Since there is no general theory available, every single equation has to be analysed separately. To obtain a detailed mathematical understanding is a challenging problem for many interesting systems of that type. The intention of the project is to study open problems in nonlinear wave equations related to singularity formation and long-time existence. The first step in studying a time evolution PDE is to establish a local well-posedness theory, i.e. one studies existence and uniqueness of solutions as well as their dependence on the data for small times. If a local solution exists there are essentially two possibilities: Either it can be continued for all times or it ceases to exist after a finite time. The first possibility is referred to as global existence while the second one is called blow up, singularity formation or simply breakdown of the solution. Interesting mathematical questions in this respect are related to the nature of the blow up and the identification of the mechanism that leads to the breakdown. Such questions are of fundamental importance since they have strong implications on the validity of the model described by the equation. The aim of the project is to study those aspects for certain wave map and Skyrme models. Wave maps are nonlinear generalizations of the well-known wave equation which extremize a geometric action. The local well- posedness theory for the wave maps model on Minkowski space is fairly well understood. However, concerning global issues there are many unanswered questions. It is known that the wave maps system from physical Minkowski space to the three-sphere exhibits self-similar solutions, i.e. solutions with smooth initial data that blow up in finite time. In order to make the problem accessible one performs an equivariant symmetry reduction and based on numerical studies, the ground state self-similar wave map is conjectured to represent the generic blow up scenario. Proving this universality of blow up is an important open issue. Furthermore, equivariant wave maps from (2+1) dimensional Minkowski space to the two-sphere show blow up via "bubbling off" of a harmonic map. However, the precise rate of blow up is only known for high equivariance indices. For low equivariance indices it is unknown whether there exists a similar stable blow up and even worse, numerics are inconclusive in this case. However, recently rigorous existence of nongeneric blow up solutions has been proved which is a very suprising feature of that system. Thus, to complete the picture, further numerical and analytical investigations are required. Finally, the project aims at initiating mathematical studies of wave maps defined on spatially compact domains as well as the Skyrme model (a generalization of the wave maps equation). Very little is known about those systems. While spatially compact wave maps are almost completely unexplored, the Skyrme model has been investigated numerically. There exists a soliton solution which is conjectured to act as a universal attractor in dynamical time evolution. The Cauchy problem for these systems deserves further investigation from both the numerical and analytical perspective.