Geometric analysis of 1+1 dimensional quasilinear wave equations

TL;DR

This paper proves global well-posedness for a class of 1+1 dimensional quasilinear wave equations with large initial data, using a geometric approach inspired by relativity and shock formation studies.

Contribution

It introduces a geometric method employing a double-null coordinate system and a quasilinear null condition to establish global solutions for large data.

Findings

Global well-posedness for large initial data.

Persistence of a quasilinear null condition.

Decoupling of wave variables from null structure equations.

Abstract

We prove global well-posedness of the initial value problem for a class of variational quasilinear wave equations, in one spatial dimension, with initial data that is not-necessarily small. Key to our argument is a form of quasilinear null condition (a "nilpotent structure") that persists for our class of equations even in the large data setting. This in particular allows us to prove global well-posedness for $C^2$ initial data of moderate decrease, provided the data is sufficiently close to that which generates a simple traveling wave. We take here a geometric approach inspired by works in mathematical relativity and recent works on shock formation for fluid systems. First we recast the equations of motion in terms of a dynamical double-null coordinate system; we show that this formulation semilinearizes our system and decouples the wave variables from the null structure equations. After solving for the wave variables in the double-null coordinate system, we next analyze the null structure equations, using the wave variables as input, to show that the dynamical coordinates are $C^1$ regular and covers the entire space-time.