Strichartz estimates for the half Klein-Gordon equation on asymptotically flat backgrounds and applications to cubic Dirac equations

TL;DR

This paper establishes endpoint Strichartz estimates for the half Klein-Gordon equation on asymptotically flat backgrounds and applies these results to prove global well-posedness and scattering for cubic Dirac equations with small initial data.

Contribution

It introduces new endpoint Strichartz estimates on asymptotically flat backgrounds and applies them to analyze cubic Dirac equations, extending Euclidean results to curved spacetimes.

Findings

Proved $L^2_t$-endpoint Strichartz estimates for Klein-Gordon equations.

Established global well-posedness and scattering for cubic Dirac equations.

Developed a parametrix construction following Metcalfe-Tataru and Xue.

Abstract

The aim of this paper is to establish the $L^2_t$-endpoint Strichartz estimate for (half) Klein-Gordon equations on a weakly asymptotically flat space-time. As an application we prove small data global well-posedness and scattering for massive cubic Dirac equations in the full subcritical range in this setting. Crucial ingredient is a parametrix contruction following the work of Metcalfe-Tataru and Xue and complements Strichartz estimates obtained by Zheng-Zhang. The proof of the global result for the cubic Dirac equation follows the strategy developed by Machihara-Nakanishi-Ozawa in the Euclidean setting.