Global smooth solutions of 2D quasilinear wave equations with higher order null conditions and short pulse initial data

TL;DR

This paper proves the global existence of smooth solutions for 2D quasilinear wave equations with higher order null conditions and short pulse initial data, addressing challenges from slow decay in 2D.

Contribution

It establishes the first global smooth solutions for a class of 2D quasilinear wave equations with higher order null conditions under short pulse data.

Findings

Global existence of smooth solutions is proven.

Key identities based on null conditions are developed.

Overcoming slow decay challenges in 2D wave equations.

Abstract

For the short pulse initial data with a first order outgoing constraint condition and optimal orders of smallness, we establish the global existence of smooth solutions to 2D quasilinear wave equations with higher order null conditions. Such kinds of wave equations include 2D relativistic membrane equations, 2D membrane equations, and some 2D quasilinear equations which come from the nonlinear Maxwell equations in electromagnetic theory or from the corresponding Lagrangian functionals as perturbations of the Lagrangian densities of linear wave operators. The main ingredients of the analysis here include looking for a new good unknown, finding some key identities based on the higher order null conditions and the resulting null frames, as well as overcoming the difficulties due to the slow decay of solutions to the 2-D wave equation, so that the solutions can be estimated precisely.