Dynamics of singularity formation for geometric wave equations

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Geometrical wave equations, such as the Wave Maps equations, the Yang-Mills equations, and nonlinear Schrödinger-type equations, come up in a variety of physical contexts. The mathematical challenge that they pose lies in analyzing qualitatively the long-time behavior of solutions for as arbitrary a class of initial data as possible. In particular, one wants to decide whether singularity formation in finite time can occur, and if so, to describe the dynamics of it. For a long time, only very special cases were amenable to a rigorous analysis. Recent advances in harmonic analysis, however, as well as new insight into how to apply methods of spectral theory in the context of nonlinear problems, have made it possible to construct examples for a greatly expanded class of situations and to uncover exciting new phenomena, for example, a great variety of different types of singularity formation. Nevertheless, a host of questions remain, and to answer some of them is the goal of this project. The topics on which the project will focus include the stability of these blow-up phenomena in the absence of any symmetry assumptions (which would make them more readily observable in computer simulations) and a useful description of the totality of all possible blow-up rates. Furthermore, as the techniques mentioned above apply only in the context of energy-critical equations, it remains to find ways of dealing with the supercritical context. This goes beyond exhibiting isolated examples (which have been known in some supercritical cases for a while). All Wave Maps equations in four-dimensional (and higher dimensional) space-time belong to the supercritical regime. This research should be of relevance to a number of other fields. Numerical simulation of the equations under investigation, simulation that is of interest to physicists, is inherently difficult to carry out accurately in the case when singularities are present. It is not uncommon that such simulations result in contradictory scenarios, from which only a careful theoretical analysis leads to correct conclusions about the underlying dynamics. On the other hand, knowledge of the correct qualitative behavior of the solutions, a goal of this project, would allow one to implement the correct computer algorithms for computing highly accurate solutions, even in the presence of singularities. A concrete example would be the propagation of lasers in weakly interacting nonlinear media. This phenomenon is described by nonlinear Schrödinger equations, and singularity formation in their solutions has a bearing on issues such as laser focusing. A second application relates to Einstein's equations of general relativity, which reduce under certain symmetry assumptions to equations of Wave Maps type. A qualitative understanding of the latter would have a bearing on our understanding of the development of black holes. Finally, the principal investigator is hopeful that his techniques will carry over to much more general classes of equations, such as parabolic nonlinear problems. These turn up in a variety of contexts (e.g., engineering problems describing processes of combustion, geometric flows).