Self-similar blowup in dispersive wave equations

The present research project is concerned with the mathematical analysis of nonlinear dispersive wave equations, a particular class of partial differential equations. Nonlinear wave equations play a central role in the description of a variety of phenomena in the natural sciences, in particular in physics, where they are fundamental for general relativity and quantum field theory. Despite this wide range of applications, the mathematical understanding of these equations is still insufficient. Consequently, dispersive wave equations are a highly active field of research in modern mathematics. The simplest physical model that is described by a wave equation is the vibrating string. Essentially, one prescribes the initial displacement as data and computes the corresponding solution of the wave equation which then yields the future development of the system. Wave equations are time evolution equations and one studies the initial value problem: Given the initial configuration at some instance of time, how does the system evolve in the future? In general, this is a very hard question since in most cases it is impossible to explicitly solve the equation. Consequently, one has to rely on indirect methods. Many nonlinear wave equations develop singularities in finite time. That is to say, the solution does not exist for all times, even if the initial data are regular. The dynamical formation of a black hole in general relativity is an example of a physical event that is described by such a breakdown. Here, the initial data are a star that is starting to collapse. Another example is a magnet whose polarization changes abruptly. In general, it is not easy to determine whether a given equation develops singularities in finite time. However, for the models considered in this research project, singularity formation can be demonstrated by the construction of explicit self-similar solutions. The natural mathematical question is then concerned with the stability of these explicit solutions. Since there are no isolated systems in nature, only stable solutions are physically relevant after all. Recently, there was spectacular progress in the study of singularity formation for wave equations. Unfortunately, most of these results deal with energy-critical and subcritical equations. From an application point of view, the challenging supercritical equations are more relevant. Consequently, the main goal of the research project is the development of novel, robust, and rigorous mathematical tools for the stability analysis of self-similar solutions to supercritical dispersive wave equations. The approach is based on a combination of methods from the classical analysis of partial differential equations, spectral theory, harmonic analysis, nonlinear functional analysis, operator theory, and numerical analysis. The long-term hope is to develop robust enough techniques so that at some point they become applicable to realistic physical systems.

The FWF project P30076 was devoted to the theoretical analysis of a class of partial differential equations describing time evolution processes in physics, biology, and geometry. Even though these equations are fundamental for many processes in the natural sciences and in pure mathematics, our mathematical understanding is embarrassingly poor. The main goal of the project was to improve this situation and to provide new insight into this important class of partial differential equations. A special focus was put on self-similar solutions which provide explicit examples of solutions that form singularities after a finite time. The physical, biological, or geometric significance of these singularities depends on their stability. In the course of the project we developed novel mathematical methods for the rigorous analysis of the stability of self-similar solutions based on spectral theory and nonlinear functional analysis. Our results advanced the field significantly and provided the ground work for attacking problems that seemed completely out of reach just a few years ago.