Proves local well-posedness for Schrödinger map flow from T^d to S^2 at σ > d/2 + 1/2 (d≥3) and to general compact Kähler N at σ > d/2 + 5/6 (d≥2), first such low-regularity result in periodic setting.
Global well-posedness of the cubic nonlinear Schr\"odinger equation on $\mathbb{T}^{2}$
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abstract
We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold in the focusing case. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular, the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. This allows to transfer the analogous results of Dodson for the non-periodic mass-critical NLS to the periodic setting. In addition, we construct an approximate periodic solution which implies sharpness of the results.
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Low-regularity Schr\"odinger map flow on high-dimensional periodic domains
Proves local well-posedness for Schrödinger map flow from T^d to S^2 at σ > d/2 + 1/2 (d≥3) and to general compact Kähler N at σ > d/2 + 5/6 (d≥2), first such low-regularity result in periodic setting.