8 research outputs found
On time periodic solutions to the conformal cubic wave equation on the Einstein cylinder
No full textInternational audienceWe consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski and Maliborski on the existence of time periodic solutions to the spherically symmetric Einstein–Klein–Gordon system, we study perturbations around the zero solution as a formal series expansion and assume that the perturbations bifurcate from one mode. In the center of this work stands a rigorous proof on how one can choose the initial data to cancel out all secular terms in the resonant system. Interestingly, our analysis reveals that the only possible choice for the existence of time periodic solutions bifurcating from the first mode is when the error terms in the expansion are all proportional to this dominant one mode. Finally, we use techniques from ordinary differential equations and establish the existence of time periodic solutions for the initial data proportional to the first mode of the linearized operator
A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes
No full textWe reconsider the unique continuation property for a general class of
tensorial Klein-Gordon equations of the form \begin{align*} \Box_{g} \phi +
\sigma \phi = \mathcal{G}(\phi,\nabla \phi) \text{,} \qquad \sigma \in
\mathbb{R} \end{align*} on a large class of asymptotically anti-de Sitter
spacetimes. In particular, we aim to generalize the previous results of
Holzegel, McGill, and the second author [14,15,24] (which established the
above-mentioned unique continuation property through novel Carleman estimates
near the conformal boundary) in the following ways: (1) We replace the
so-called null convexity criterion (the key geometric assumption on the
conformal boundary needed in [24] to establish the unique continuation
properties) by a more general criterion that is also gauge invariant. (2) Our
new unique continuation property can be applied from a larger, more general
class of domains on the conformal boundary. (3) Similar to [24], we connect the
failure of our generalized null convexity criterion to the existence of certain
null geodesics near the conformal boundary. These geodesics can be used to
construct counterexamples to unique continuation. Finally, our gauge-invariant
criterion and Carleman estimate will constitute a key ingredient in proving
unique continuation results for the full nonlinear Einstein-vacuum equations,
which will be addressed in a forthcoming paper of Holzegel and the second
author [16].Comment: 46 pages, 5 figures, minor changes, added references. Matches the
published versio
On the stability of the blow-up for the wave maps and the cubic wave equation
No full textIn Thesis is divided into two parts. In the first part, we study the stability of the blow-up for the wave maps equation whereas in the second part we address the stability of the blowup for the cubic wave equation in higher space dimmensions
On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants
No full textWe consider the -dimensional Einstein equations with negative
cosmological constant coupled to a spherically-symmetric, massless scalar field
and study perturbations around the Anti-de Sitter spacetime. We derive the
resonant systems, pick out vanishing secular terms and discuss issues related
to small divisors. Most importantly, we rigorously establish (sharp, in most of
the cases) asymptotic behaviour for all the interaction coefficients. The
latter is based on uniform estimates for the eigenfunctions associated to the
linearized operator as well as on some oscillatory integrals.Comment: Minor improvements to match the published version, will appear in
Annales Henri Poincar\'
Non-linear periodic waves on the Einstein cylinder
No full textMotivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder . For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the -sphere. In all three cases, the equations reduce to semi-linear wave equations. Our proof relies on a theorem of Bambusi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the non-degeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is nonresonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to backreaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting.90 pages, 10 figures, Mathematica files added as ancillar
The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
No full textIn this paper, we consider vacuum asymptotically anti-de Sitter spacetimes with conformal boundary . We
establish a correspondence, near , between such spacetimes and
their conformal boundary data on . More specifically, given a
domain , we prove that the coefficients
and (the undetermined
term or stress energy tensor) in a Fefferman-Graham expansion of the metric
from the boundary uniquely determine near , provided
satisfies a generalised null convexity condition (GNCC). The GNCC
is a conformally invariant criterion on , first identified by
Chatzikaleas and the second author, that ensures a foliation of pseudoconvex
hypersurfaces in near , and with the pseudoconvexity
degenerating in the limit at . As a corollary of this result, we
deduce that conformal symmetries of on domains satisfying the GNCC extend to
spacetimes symmetries near . The proof, which does not require any
analyticity assumptions, relies on three key ingredients: (1) a calculus of
vertical tensor-fields developed for this setting; (2) a novel system of
transport and wave equations for differences of metric and curvature
quantities; and (3) recently established Carleman estimates for tensorial wave
equations near the conformal boundary.Comment: 60 pages, 1 figure. Version accepted at ARM
