Gradient NLW on curved background in 4+1 dimensions

TL;DR

This paper establishes a precise local well-posedness result for the Gradient Nonlinear Wave Equation on nonsmooth curved backgrounds in 4+1 dimensions, introducing new analytical tools and estimates.

Contribution

It introduces variable coefficient Bourgain's $X^{s,b}$ spaces and a trilinear wave packet decomposition to handle the equation on nonsmooth backgrounds.

Findings

Proved sharp local well-posedness for the equation.

Developed variable coefficient $X^{s,b}$ spaces.

Established a key trilinear estimate.

Abstract

We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate.