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

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 $\varepsilon \rightarrow 0$.