Conditional global regularity of Schr\"odinger maps: sub-threshold dispersed energy

TL;DR

This paper proves global regularity for Schr"odinger maps in 2+1 dimensions with subthreshold initial data, assuming energy dispersion and boundedness conditions, by establishing improved estimates and adapting existing techniques.

Contribution

It introduces a novel approach to prove global regularity for Schr"odinger maps under subthreshold energy dispersion, extending local smoothing and bilinear estimates to the nonlinear setting.

Findings

Global smooth solutions exist under energy dispersion assumptions.

Established improved local smoothing and bilinear Strichartz estimates.

Derived global-in-time bounds on Sobolev norms of solutions.

Abstract

We consider the Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger maps.