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

TL;DR

This paper proves the global existence and scattering of small smooth solutions for a class of quasilinear wave systems on a mixed space, using transformations and advanced estimates, including the vector-field method and ghost weight technique.

Contribution

It introduces a method to analyze quasilinear wave systems on $ ^2 imes $ by transforming the equations and applying novel weighted estimates, extending previous results to more complex systems.

Findings

Global existence of small data solutions established

Scattering behavior of solutions demonstrated

Applicable to various physical wave models

Abstract

In this paper, we are concerned with the global existence and scattering of small data smooth solutions to a class of quasilinear wave systems on the product space $\mathbb{R}^2\times\mathbb{T}$. These quasilinear wave systems include 3D irrotational potential flow equation of Chaplygin gases, 3D relativistic membrane equation, some 3D quasilinear wave equations which come from the corresponding Lagrangian functionals as perturbations of the Lagrangian densities of linear waves, and nonlinear wave maps system. Through looking for some suitable transformations of unknown functions, the nonlinear wave system can be reduced into a more tractable form. Subsequently, by applying the vector-field method together with the ghost weight technique as well as deriving some kinds of weighted $L^\infty-L^\infty$ and $L^\infty-L^2$ estimates of solution $w$ to the 2D linear wave equation $\Box w=f(t,x)$, the global existence and scattering of small data solutions are established.