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

TL;DR

The paper develops a semiglobal scattering theory for nonlinear perturbations of the Minkowskian wave equation in a neighborhood of spacelike infinity, incorporating past and future null infinity with data posed on an ingoing null cone and on I^-. It establishes weighted energy estimates, commutation-aligned arguments, and time-inversion techniques to propagate polyhomogeneous expansions from past to future null infinity, and it provides an algorithmic method to compute expansion coefficients. The framework handles short-range and linear long-range perturbations, extends to systems like Einstein vacuum equations in harmonic gauge, and clarifies the limitations of peeling in various dimensions. By summing over spherical harmonics, the work connects fixed-mode asymptotics to global polyhomogeneous behavior, enabling comprehensive asymptotic descriptions and new insights into the smoothness of null infinity and gravitational radiation. The results yield robust scattering theory beyond finite energy, including polyhomogeneous data and slowly decaying scenarios, with broad applicability to perturbations of wave equations and to general relativity in harmonic gauge.

Abstract

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].

Figures (7)

• Figure 1: Our scattering data are given on a null cone $\underline{\mathcal{C}}_{v_0}$ and on $\mathcal{I}^-_{v\geq v_0}$. Depicted is the Penrose diagram of $\underline{\mathcal{D}}$, the domain of dependence of $\underline{\mathcal{C}}_{v_0}\cup\mathcal{I}^-_{v\geq v_0}.$ The darker shaded region corresponds to an initial data slab and is only referred to in \ref{['sec:EVE']}.
• Figure 2: The two different compactifications $\mathcal{D}$ and $\boldsymbol{\mathcal{D}}$. The latter compactification is specifically tailored to capturing the no incoming radiation condition. The boundary $\boldsymbol{I}^0$ corresponds to limit points $\boldsymbol{\rho}_0\to0$ (which includes both $\mathcal{I}^-$ and $I^0$ in the $\mathcal{D}$-picture). Solutions with (without) incoming radiation are regular w.r.t. smooth vector fields that are tangent to the boundaries of $\mathcal{D}$ ($\boldsymbol{\mathcal{D}}$).
• Figure 3: A schematic guide to reading the notation $\psi\in\mathcal{A}_{\mathrm{b,b,phg}}^{a_-,a_0,\overline{(0,0)}}(\mathcal{D})$. In case $\psi$ is polyhomogeneous towards all boundaries, we simply write $\mathcal{A}_{\mathrm{phg}}$ instead of $\mathcal{A}_{\mathrm{phg,phg,phg}}$; in case $\psi$ is merely bounded towards all boundaries, we write $H_{\mathrm{b}}$ rather than $\mathcal{A}_{\mathrm{b,b,b}}$. In contrast, the number of superscripts always matches the number of boundaries. We also notice that, if we work on a space with only two boundaries, e.g. $\mathcal{D}\cap\{t\leq 0\}$ or $\boldsymbol{\mathcal{D}}$, then we correspondingly only write two sub- and superscripts (e.g. $\mathcal{A}_{\mathrm{phg}}^{\mathcal{E}_0,\mathcal{E}_+}(\boldsymbol{\mathcal{D}})$).
• Figure 4: Depicted are the integral curves of the scaling vector field. The shaded regions are $\mathcal{D}^+,\mathcal{D}^-$, see \ref{['fig:D-a']}.
• Figure 5: Depiction of the regions $\underline{\mathcal{D}}^{\pm}$ in a Penrose diagram and with respect to the compactification $\mathcal{D}$.

Theorems & Definitions (310)

• Theorem 1.1: Main result, prosaic version
• Theorem 1.2: from kehrberger_case_2022-1
• Theorem 1.3
• Remark 1.1
• Conjecture 1.1
• Remark 1.2: Initial data assumptions: Conormality vs. Conformal Smoothness
• Remark 1.3: Expanding around Minkowski
• Remark 1.4: Linear long-range potentials
• Remark 1.5: Estimate on the error-term
• Definition 1.1