Geometric analysis of biwave maps

Wave equations can be successfully applied to describe many physical phenomena in nature. For example, linear wave equations of second order may be used to model the vibration of a string or the propagation of electromagnetic waves in vacuum. The fact that the wave equation is linear implies that its solutions exist for all times, such kinds of solutions are called global. On the other hand, if one considers a non-linear wave equation, then its solutions may form a singularity after finite time. One prominent example in this regard are the Einstein equations which describe the dynamics of our universe. In general, the solutions of the Einstein equations will develop a singularity after finite time, which manifests itself in the existence of black holes. Another prominent non-linear wave equation of second order, which already received a lot of attention, is the so-called wave maps equation. This equation has multiple applications in theoretical physics, i. e. in quantum field theory, where it is used to describe the dynamics of certain particles. An interesting property of the wave maps equation is the fact that it admits both global solutions but also solutions which develop a singularity after finite time. This project will investigate a non-linear wave equation of fourth order, which generalizes the wave maps equation, the so-called biwave maps equation. Biwave maps have important applications in elasticity theory where they can be used to model the dynamics of an elastic material. For example, if one thinks of a membrane made of rubber which is fixed at its ends, then the biwave maps equation can give information on the dynamics of the membrane if one acts with an external force on it. Hence, it is evident that the biwave maps equation depends heavily on the geometry of the membrane. A particular difficulty in the mathematical analysis of biwave maps stems from the fact that every wave map is also a solution of the biwave maps equation. However, it is to be expected that there also exist additional solutions, which are not wave maps, and it is one central aim of this project to identify the latter. The central questions of this project can be summarized as follows. 1)Which are the basic mathematical properties of the biwave maps equation and its solutions? 2)What is the mathematical relation between wave and biwave maps? 3)Under which conditions (i.e. on the geometry) does there exist a solution of the biwave maps equation for all times? Are there also cases in which we can even explicitly solve the equation for biwave maps? 4)When does the biwave maps equation develop a singularity and what are the properties of such a singularity? It is to be expected that this project bears interesting new mathematical structures on non-linear wave equations of higher order which will also shed new light on wave maps.