On wave systems with antisymmetric potential in dimension d >= 4 and well-posedness for (half-)wave maps

TL;DR

This paper establishes a priori estimates for certain wave systems with antisymmetric potentials in dimensions four and higher, leading to well-posedness results for wave and half-wave maps with small initial data.

Contribution

It introduces new a priori estimates for wave systems with antisymmetric potentials applicable to wave and half-wave maps, proving well-posedness in high dimensions.

Findings

A priori estimates for wave systems with antisymmetric potentials

Well-posedness for wave and half-wave maps in high dimensions

Applicability of assumptions to specific geometric wave equations

Abstract

We prove a priori estimates for wave systems of the type \[ \partial_{tt} u - \Delta u = \Omega \cdot du + F(u) \quad \text{in $\mathbb{R}^d \times \mathbb{R}$} \] where $d \geq 4$ and $\Omega$ is a suitable antisymmetric potential. We show that the assumptions on $\Omega$ are applicable to wave- and half-wave maps, the latter by means of the Krieger-Sire reduction. We thus obtain well-posedness of those equations for small initial data in $\dot{H}^{\frac{d}{2}}(\mathbb{R}^d)$.