Stable blowup in supercritical wave equations

Partial differential equations are a fundamental mathematical tool for the description of dynamical processes. Since Newton the physical laws are formulated in the language of differential equations. A simple example is the description of the vibrating string by the wave equation. One prescribes the initial state of the string (e.g. the displacement) and the solution of the differential equation yields the future behavior of the system. However, only for very simple differential equations it is possible to compute an explicit solution. In fact, in many cases it is not even clear upfront that solutions exist at all. It is therefore a task for mathematicians to develop suitable theories for the solvability of differential equations. Moreover, in complex systems it is common that solutions exist for small times but break down afterwards. A physical example is the formation of a black hole where a singularity in spacetime occurs. The formation of the latter is signalled by the breakdown of the solution of a certain differential equation. It is thus important to understand possible mechanisms that cause the breakdown of solutions. Moreover, it is necessary to understand the features of the solution shortly before the breakdown. The goal of the project is to study such questions for a class of wave equations that originate from geometry and/or physics. The mathematical understanding of singularity formation for these equations is still embarrasingly poor and the project strives to substantially advance our knowledge in this field.

The research project dealt with the formation of singularities in nonlinear wave equations. These types of equations describe a variety of fundamental physical processes. An abrupt change in the physical system, such as the spontaneous reversal of a magnet or the formation of a black hole in general relativity, is mathematically expressed by the formation of a so-called singularity. A precise mathematical understanding of singularities is therefore of great interest, and the aim of the project was to improve this understanding or, in many cases, to develop it in the first place. Within the framework of the project, it was possible to develop a new theory of the stability of singularities in general coordinate systems and to understand singularities under minimal regularity conditions. These results relate to simplified models, but the mechanisms discovered in the process are also active in realistic physical systems.