Global solutions for 1D cubic defocusing dispersive equations, Part V: low regularity NLS

Abstract

This article is motivated by a broad conjecture, formulated by the first and last authors in earlier work, asserting that one-dimensional cubic defocusing dispersive flows with small initial data have global, dispersive solutions. The conjecture was first established for a class of semilinear Schr\"odinger-type models at $L^2$ regularity, the classical cubic NLS among them. In a complementary direction, Harrop-Griffiths, Killip and Vi\c{s}an have recently shown, using the completely integrable structure, that the cubic NLS is globally well-posed in $H^s$ for every $-\tfrac12 < s < 0$. Our aim here is to extend the reach of the global well-posedness conjecture for one dimensional cubic NLS problems to data which is small in negative Sobolev spaces, and to show that global dispersive bounds persist there. We do so for a broad class of nonlinearities which includes the cubic NLS but which in general generates flows that are not completely integrable. Our method is correspondingly robust, resting on density-flux identities, interaction Morawetz estimates and an implicit normal form transformation rather than on integrability, and it reaches all the way to the scaling-critical threshold, namely $s > -\tfrac12$. As in the earlier work, the global bounds we obtain include both $L^6_{t,x}$ Strichartz estimates and bilinear $L^2_{t,x}$ estimates; these are new even for the classical defocusing cubic NLS at negative Sobolev regularity. There, by scaling, our dispersive bounds also extend to the large data case.