Global well-posedness for Schrodinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces

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Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, background): “in [104], where the identical statement is proved in the context of low regularities.”
Existence of global solutions to the derivative NLS equation with the inverse scattering transform method (2016, methodology): “Global existence under the constraint (1.2) was established in Hs(R) with subsequent generations of the Fourier transform restriction method and the so-called Imethod, e.g., for s > 3233 by Takaoka [31], for s > 2 3 and s > 1 2 by Colliander et al. [7] and [8] respectively, and finally for s = 12 by”
INVARIANT WEIGHTED WIENER MEASURES AND ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVE NLS. (2010, methodology): “Then, Colliander–Keel–Staffilani– Takaoka and Tao [15, 16] established global well-posedness (GWP) for data in H σ (R), σ > 1/2, of small L2 norm using the so-called I-method on the gauge equivalent equation (see also [41]).”