Studies of singularity formation in gravity

An important feature of many nonlinear evolution equations is that their solutions, corresponding to smooth initial conditions, may form singularities in a finite time. Such a phenomenon is called ``blow-up`` and has been a subject of intensive studies in many fields ranging from fluid dynamics to general relativity. Given a nonlinear evolution equation, the key question: `Can blow-up occur for some initial conditions?` is often difficult. Two famous examples for which the answer is not known are the Navier-Stokes equation and the Einstein equation. When a particular equation allows finite time singularities many related questions come up: When and where does the blow-up occur? What is the character of blow-up? Can a solution be continued past the singularity? Insight into blow-up phenomena has been gained by studies of critical behavior. For systems with only two possible end-states (e.g. singularity formation and dispersion) fine-tuning the data to the boundary between the end-states results in the evolution approaching the singularity in a universal way, independent of the special form of initial data. In gravity critical collapse occurs when the gravitational attracting and the repulsive forces of matter almost balance. The systematic studies of critical gravitational collapse began in the early nineties based on the analytical work of Christodoulou and the numerical work by Choptuik. Although these phenomena have now been observed numerically for many evolutionary equations, their mathematical understanding remains a challenge. It is the aim of this project, by combining numerical and analytical methods to study mathematical and physical properties of singularity formation for the following models: i) The spherically-symmetric, equivariant SU(2) sigma-model on Minkowski background. Analytically we intend to study existence and uniqueness of solutions of the linearized problem, stability and instability of solutions which are expected to act as attractors, spectral properties of the linearized perturbation operator of special solutions, existence of attractors in the nonlinear system. ii)Wave maps in the critical dimension 2+1 coupled to gravity with negative cosmological constant. Numerically we would like to analyse critical collapse at the threshold of black hole formation. Is the static soliton which exists for zero cosmological constant of importance in this setting? iii)For the Einstein-massless scalar field in 3+1 dimensions with spherical symmetry we intend to study global aspects of critical collapse. In particular, we would like to focus on the issue of how much of the backscattered radiation eventually falls into the black hole.

An important feature of many nonlinear evolution equations is that their solutions, corresponding to smooth initial conditions, may form singularities in a finite time. Such a phenomenon is called ``blow-up`` and has been a subject of intensive studies in many fields ranging from fluid dynamics to general relativity. Given a nonlinear evolution equation, the key question: `Can blow-up occur for some initial conditions?` is often difficult. Two famous examples for which the answer is not known are the Navier-Stokes equation and the Einstein equation. When a particular equation allows finite time singularities many related questions come up: When and where does the blow-up occur? What is the character of blow-up? Can a solution be continued past the singularity? Insight into blow-up phenomena has been gained by studies of critical behavior. For systems with only two possible end-states (e.g. singularity formation and dispersion) fine-tuning the data to the boundary between the end-states results in the evolution approaching the singularity in a universal way, independent of the special form of initial data. In gravity critical collapse occurs when the gravitational attracting and the repulsive forces of matter almost balance. The systematic studies of critical gravitational collapse began in the early nineties based on the analytical work of Christodoulou and the numerical work by Choptuik. Although these phenomena have now been observed numerically for many evolutionary equations, their mathematical understanding remains a challenge. It is the aim of this project, by combining numerical and analytical methods to study mathematical and physical properties of singularity formation for the following models: 1. The spherically-symmetric, equivariant SU(2) sigma-model on Minkowski background. Analytically we intend to study existence and uniqueness of solutions of the linearized problem, stability and instability of solutions which are expected to act as attractors, spectral properties of the linearized perturbation operator of special solutions, existence of attractors in the nonlinear system. 2. Wave maps in the critical dimension 2+1 coupled to gravity with negative cosmological constant. Numerically we would like to analyse critical collapse at the threshold of black hole formation. Is the static soliton which exists for zero cosmological constant of importance in this setting? 3. For the Einstein-massless scalar field in 3+1 dimensions with spherical symmetry we intend to study global aspects of critical collapse. In particular, we would like to focus on the issue of how much of the backscattered radiation eventually falls into the black hole.