Spectral Properties and Stability of Self-Similar Wave Maps

In this thesis the Cauchy problem and in particular the question of singularity formation for co--rotational wave maps from 3+1 Minkowski space to the three--sphere $S^3$ is studied. Numerics indicate that self--similar solutions of this model play a crucial role in dynamical time evolution. In particular, it is conjectured that a certain solution $f_0$ defines a universal blow up pattern in the sense that the future development of a large set of generic blow up initial data approaches $f_0$ . Thus, singularity formation is closely related to stability properties of self--similar solutions. In this work, the problem of linear stability is studied by functional analytic methods. In particular, a complete spectral analysis of the perturbation operators is given and well--posedness of the linearized Cauchy problem is proved by means of semigroup theory and, alternatively, the functional calculus for self--adjoint operators. These results lead to growth estimates which provide information on the stability of self--similar wave maps. Finally, convergence properties of $f_n$ for large $n$ and the spectra of the corresponding perturbation operators are investigated. The thesis is intended to be self--contained as far as possible, i.e. all the mathematical requirements are carefully introduced, including proofs for many results which could be found elsewhere.

Spectral Properties And Linear Stability Of Self-Similar Wave Maps

Roland Donninger

We study co-rotational wave maps from (3 + 1)-Minkowski space to the three-sphere S-3. It is known that there exists a countable family {f(n)} of self-similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well-posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from f(n) by letting n -> infinity.

2009

Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow

Willie Wai Yeung Wong

We study constant mean curvature Lorentzian hypersurfaces of R^{1,d+1} from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric solutions exhibit one of three (future) asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like cylinder isometric to some RxS^d and (iii) infinite expansion to the future converging asymptotically to a time translation of the de Sitter solution. For class (iii) we examine the future stability properties of the solutions under arbitrary (not necessarily spherically symmetric) perturbations. We show that the usual notions of asymptotic stability and modulational stability cannot apply, and connect this to the presence of cosmological horizons in these class (iii) solutions. We can nevertheless show the global existence and future stability for small perturbations of class (iii) solutions under a notion of stability that naturally takes into account the presence of cosmological horizons. The proof is based on the vector field method, but requires additional geometric insight. In particular we introduce two new tools: an inverse-Gauss-map gauge to deal with the problem of cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson tensor to obtain natural energy quantities.

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Roland Donninger

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as

t^{-2\ell-3}

for

t \to \infty

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

t^{-2\ell-4}

. We give a proof of

t^{-2\ell-2}

decay for general data in the form of weighted

L^1

L^\infty

bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain

t^{-2\ell-3}

. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

2011

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Roland Donninger

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as

t^{-2\ell-3}

for

t \to \infty

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

t^{-2\ell-4}

. We give a proof of

t^{-2\ell-2}

decay for general data in the form of weighted

L^1

L^\infty

bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain

t^{-2\ell-3}

2011