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

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: $$ \partial_u(r^{-2\ell}\partial_v(r^2\partial_v)^{\ell}(r\phi_{\ell}))=0, $$ which are satisfied if $\Box_\eta\phi=0$. 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].