An Approximation Scheme for Schrödinger Maps

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On Schrödinger maps (2008, methodology): “For the corresponding equation in the case of compact Kähler manifolds smooth solutions were constructed by Ding and Wang, see [5], and, more recently, McGahagan obtained the best result up to date, see [13].”
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 (2007, background): “It is known that sufficiently smooth solutions exist locally in time, even for large data (see, for example, [19], [2], [3], [13], [9] and the references therein).”
On the global well-posedness of the one-dimensional Schrodinger map flow (2008, background): “Theorem 1.1. (See [DW2, M2 ].) Let (M m , g) be a complete Riemannian manifold”
Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “Our starting point is the local existence and uniqueness of smooth solutions of the Schrödinger map equation (see, for example, [32]): if φ0 ∈ H∞ Q , then there is T = T (‖φ0‖H2d+20 Q ) > 0 and a unique solution φ ∈ C((−T, T ) : H ∞ Q ) of the initial value problem (1.”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, background): “It is known that sufficiently smooth solutions exist locally in time, even for large data (see, for example, [3,5,16,22] and the references therein).”