Global Well-Posedness For Half-Wave Maps With $S^2$ and $\mathbb{H}^2$ Targets For Small Smooth Initial Data

TL;DR

This paper establishes the global well-posedness of half-wave maps targeting spheres and hyperbolic planes for small initial data in specific Sobolev and Besov spaces, advancing understanding of these geometric PDEs.

Contribution

It proves the first global well-posedness results for half-wave maps with $S^2$ and $ ext{H}^2$ targets under small initial data conditions.

Findings

Global well-posedness for $S^2$ target with small initial data in $ ext{dot}H^{n/2} \times \text{dot}H^{n/2-1}$.

Global well-posedness for $ ext{H}^2$ target with small smooth initial data in Besov spaces.

Extension of well-posedness theory to half-wave maps with non-compact target spaces.

Abstract

We prove global well-posedness for the half-wave map with $S^2$ target for small $\dot{H}^{\frac{n}{2}} \times \dot{H}^{\frac{n}{2}-1}$ initial data. We also prove the global well-posedness for the equation with $\mathbb{H}^2$ target for small smooth $\dot{B}^{\frac{n}{2}}_{2,1} \times \dot{B}^{\frac{n}{2}-1}_{2,1}$ initial data.