On optimal blowup stability for supercritical wave equations

Partial differential equations (equations involving derivatives in several variables) play a fundamental role in all natural sciences, especially in physics. They are essential building blocks for modeling various phenomena, and appear prominently in many areas such as quantum physics, theory of relativity, electrodynamics, and fluid mechanics, to name some. The equations studied in this research project are among the prototypical wave equations and have their origins in physics. The first equations of this type were already investigated in the 18th century by d`Alembert. He studied what is today referred to as the one-dimensional free wave equation, as a model of a vibrating string. His findings are still taught in introductory classes on partial differential equations. Over the years, the multi-dimensional variants of the free wave equation were also investigated and are now very well understood. In today`s research, mainly non-linear versions and other more involved but related equations are studied. These often model complicated physical events and the associated solutions exhibit far more complex and interesting dynamics. One such feature that solutions can exhibit, is the collapse in finite time. Such solutions are usually referred to as blowups and indicate a significant change in the underlying system (e.g. the implosion of a star). In addition, they often play an important role in the evolution of generic solutions. Therefore, the quest for a better understanding of the behavior of solutions to such equations naturally leads to a detailed analysis of such blowups. The focus of this project is on two special wave equations, namely the wave maps equation and the focusing wave equation with a power nonlinearity. Both equations have explicitly known blowups which have already been shown in several papers to play an essential role in the behavior of generic solutions. In the course of this project, the necessary methods will be developed to prove this influence for the largest possible set of initial data, as well as randomized initial data. Tools from different areas of mathematics, such as the theory of ordinary differential equations, functional analysis, and harmonic analysis, will be used. Furthermore, standard tools from the theory of partial differential equations will also be employed and further developed.

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