Conditional global regularity of Schrödinger maps: sub-threshold dispersed energy

We consider the Schrödinger map initial value problem into the sphere in 2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a priori $L^4$ boundedness condition on the solution, we prove that the Schrödinger map system admits a unique global smooth solution provided that the initial data is sufficiently energy-dispersed. Also shown are global-in-time bounds on certain Sobolev norms of the solution. Toward these ends we establish improved local smoothing and bilinear Strichartz estimates, adapting the Planchon-Vega approach to such estimates to the nonlinear setting of Schrödinger maps.

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