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 $\epsilon \to 0$.