Geometric analysis of $1+1$ dimensional quasilinear wave 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 wellposedness 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 spacetime.