arXiv subjects

Lionor Kehrberger

Publications and source records attributed to Lionor Kehrberger.

5 recordsLinked to original sources

arXiv2026

A note on exterior stability of isolated singularity formation for nonlinear wave equations

We study the stability of the exterior of Type I and Type II singularity formation for the wave maps equation in $\mathbb{R}^{d+1}$ with $d\geq2$ and the power nonlinear wave equation in $\mathbb{R}^{d+1}$ with $d\geq3$:Given characteristic initial data on the backwards lightcone of the singularity $\mathcal{C}=\{t+r=0\}$ converging to the singular background solution along with suitable data on an outgoing cone, we establish existence in a region $\{t+r\in(0,v_1),t-r\in(-1,0)\}$ for some suitably small $v_1$, i.e. all the way to the Cauchy horizon. Our result hinges on a particular set of assumptions on the regularity properties of these initial data, which conjecturally can be recovered by a more detailed stability analysis of the behaviour inside the past light cone; indeed, in certain settings, this was achieved in [BDS21,KAD26], and we strongly expect they can be proved in many other settings as well. The proof goes via a suitable change of coordinates and an application of the scattering result of [KK25], which, in particular, also applies to scaling-critical potentials. While no symmetry assumption is made for the power nonlinear wave equation, we only provide the proof in the corotational symmetry class for the wave maps equation, but we also sketch how to lift this restriction.

math.AP↗

arXiv2025

Scattering, Polyhomogeneity and Asymptotics for Quasilinear Wave Equations From Past to Future Null Infinity

We present a general construction of semiglobal scattering solutions to quasilinear wave equations in a neighbourhood of spacelike infinity including past and future null infinity, where the scattering data are posed on an ingoing null cone and along past null infinity. More precisely, we prove weighted, optimal-in-decay energy estimates and propagation of polyhomogeneity statements from past to future null infinity for these solutions, we provide an algorithmic procedure how to compute the precise coefficients in the arising polyhomogeneous expansions, and we apply this procedure to various examples. As a corollary, our results directly imply the summability in the spherical harmonic number $\ell$ of the estimates proved for fixed spherical harmonic modes in the papers [Keh22b,KM24] from the series "The Case Against Smooth Null Infinity". Our (physical space) methods are based on weighted energy estimates near spacelike infinity similar to those of [HV23], commutations with (modified) scaling vector fields to remove leading order terms in the relevant expansions, time inversions, as well as the Minkowskian conservation laws: $$ \partial_u(r^{-2\ell}\partial_v(r^2\partial_v)^{\ell}(rϕ_{\ell}))=0, $$ which are satisfied if $\Box_ηϕ=0$. Our scattering constructions apply to systems of equations as well and go beyond the usual class of finite energy solutions. We use this to also derive a scattering theory and prove propagation of polyhomogeneity for the Einstein vacuum equations in a harmonic gauge. In the process, we also need to introduce a novel ansatz accounting for the stronger-than-Schwarzschildean divergence of the light cones, which, in particular, extends existing exterior stability of Minkowski statements in harmonic gauge to allow for slowly decaying data as considered in [Bie10].

math.AP↗

arXiv2025

Linear and nonlinear late-time tails on dynamical black hole spacetimes via time integrals

We prove the global leading-order late-time asymptotic behaviour of solutions to inhomogeneous wave equations on dynamical black hole exterior backgrounds that settle down to Schwarzschild backgrounds with arbitrarily small decay rates. In particular, we show that for non-spherically symmetric solutions arising from compactly supported initial data, the late-time decay deviates from Price's law -- governing the decay for stationary black hole backgrounds -- by exhibiting slower time decay by exactly one additional power. Our proof is based around the observation that the emergence of late-time "tails", featuring inverse-polynomial decay in time, is intimately connected to conformal irregularity properties in space (towards future null infinity) of time integrals of the solutions. This relationship is exploited through a purely physical-space approach based around energy estimates, in which the dynamical wave operator is treated as a Schwarzschild wave operator with an inhomogeneous term. Going from almost-sharp decay to global asymptotics is achieved by exploiting this relation for the difference between the solution and a carefully chosen global tail function. We further apply our method to several examples of nonlinear wave equations and comment on its robustness to more general settings, such as dynamical spacetimes converging to sub-extremal Kerr spacetimes and higher-dimensional wave operators with even or odd spacetime dimensions.

gr-qc↗

arXiv2024

The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes

Starting from Post-Newtonian predictions for a system of $N$ infalling masses from the infinite past, we formulate and solve a scattering problem for the system of linearised gravity around Schwarzschild as introduced in [DHR19]. The scattering data are posed on a null hypersurface $\mathcal C$ emanating from a section of past null infinity $\mathcal I^-$, and on the part of $\mathcal I^-$ that lies to the future of this section: Along $\mathcal C$, we implement the Post-Newtonian theory-inspired hypothesis that the gauge-invariant components of the Weyl tensor $α$ and $\underlineα$ (a.k.a. $Ψ_0$ and $Ψ_4$) decay like $r^{-3}$, $r^{-4}$, respectively, and we exclude incoming radiation from $\mathcal I^-$ by demanding the News function to vanish along $\mathcal I^-$. We also show that compactly supported gravitational perturbations along $\mathcal I^-$ induce very similar data, with $α$, $\underlineα$ decaying like $r^{-3}$, $r^{-5}$ along $\mathcal C$. After constructing the unique solution to this scattering problem, we provide a complete analysis of the asymptotic behaviour of projections onto fixed spherical harmonic number $\ell$ near spacelike $i^0$ and future null infinity $\mathcal I^+$. Using our results, we also give constructive corrections to popular historical notions of asymptotic flatness such as Bondi coordinates or asymptotic simplicity. In particular, confirming earlier heuristics due to Damour and Christodoulou, we find that the peeling property is violated both near $\mathcal I^-$ and near $\mathcal I^+$, with e.g. $α$ near $\mathcal I^+$ only decaying like $r^{-4}$ instead of $r^{-5}$. We also find that the resulting solution decays slower towards $i^0$ than often assumed, with $α$ decaying like $r^{-3}$ towards $i^0$. The issue of summing up the fixed angular mode estimates in $\ell$ is dealt with in forthcoming work.

gr-qc↗

arXiv2024

The Case Against Smooth Null Infinity IV: Linearised Gravity Around Schwarzschild -- An Overview

This paper is the fourth in a series dedicated to the mathematically rigorous asymptotic analysis of gravitational radiation under astrophysically realistic setups. It provides an overview of the physical ideas involved in setting up the mathematical problem, the mathematical challenges that need to be overcome once the problem is posed, as well as the main new results we obtain in the companion paper [KM24]. From the physical perspective, this includes a discussion of how Post-Newtonian theory provides a prediction on the gravitational radiation emitted by $N$ infalling masses from the infinite past in the intermediate zone, i.e. up to some finite advanced time. From the mathematical perspective, we then take this prediction, together with the condition that there be no incoming radiation from $\mathcal{I}^-$, as a starting point to set up a scattering problem for the linearised Einstein vacuum equations around Schwarzschild and near spacelike infinity, and we outline how to solve this scattering problem and obtain the asymptotic properties of the scattering solution near $i^0$ and $\mathcal{I}^+$. The full mathematical details are presented in the companion paper [KM24].

gr-qc↗