Global existence and scattering of small data smooth solutions to quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$, II

TL;DR

This paper extends previous work to prove global existence and scattering of small smooth solutions for a broader class of quasilinear wave systems on , combining null condition criteria.

Contribution

It advances the theory by establishing global well-posedness for general 3-D quadratically quasilinear wave systems under null conditions.

Findings

Proved global existence for remaining nonlinearities.

Unified results for a class of wave systems on .

Validated null condition sufficiency for global solutions.

Abstract

In our previous paper [Fei Hou, Fei Tao, Huicheng Yin, Global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$, Preprint (2024), arXiv:2405.03242], for the $Q_0$-type quadratic nonlinearities, we have shown the global well-posedness and scattering properties of small data smooth solutions to the quasilinear wave systems on $\mathbb{R}^2\times\mathbb{T}$. In this paper, we start to solve the global existence problem for the remaining $Q_{\alpha\beta}$-type nonlinearities. By combining these results, we have established the global well-posedness of small solutions on $\mathbb{R}^2\times\mathbb{T}$ for the general 3-D quadratically quasilinear wave systems when the related 2-D null conditions are fulfilled.