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We prove global well-posedness for a cubic, non-local Schr\"odinger equationwith radially-symmetric initial data in the critical space $L^2(\R^2)$, usingthe framework of Kenig-Merle and Killip-Tao-Visan. As a consequence, we obtaina global well-posedness result for Schr\"odinger maps from $\R^2$ into $\S^2$(Landau-Lifshitz equation) with radially symmetric initial data (with no sizerestriction).

Global well-posedness for 2D radial Schr\"odinger maps into the sphere