Global Weak Solutions for the Half-Wave Maps Equation in $\mathbb{R}$

TL;DR

This paper proves the existence of global weak solutions for the half-wave maps equation on the real line with large initial data, using a regularization approach and weak convergence methods.

Contribution

It establishes the first existence results for weak solutions of the half-wave maps equation with large data in critical Sobolev spaces.

Findings

Existence of weak global solutions for large initial data.

Convergence of regularized solutions to weak solutions.

Methodology applicable to similar nonlocal geometric PDEs.

Abstract

We establish the existence of weak global solutions of the half-wave maps equation with the target $S^2$ on $\mathbb{R}^{1+1}$ with large initial data in $\dot{H}^1 \cap \dot{H}^{\frac{1}{2}}(\mathbb{R})$. We first prove the global well-posedness of a regularized equation. Then we show that the weak limit of the regularized solutions is a weak solution of the half-wave maps equation as the regularization parameter $\varepsilon \rightarrow 0$.