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We prove that the cubic nonlinear Schr\"odinger equation (both focusing anddefocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity$s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], butnot previously for any $s<0$. The scaling-critical value $s=-\frac12$ isnecessarily excluded here, since instantaneous norm inflation is known to occur[11, 40, 48].  We also prove (in a parallel fashion) well-posedness of the real- andcomplex-valued modified Korteweg-de Vries equations in $H^s(\mathbb R)$ for any$s>-\frac12$. The best regularity achieved previously was $s\geq \frac14$; see[15, 24, 33, 39].  To overcome the failure of uniform continuity of the data-to-solution map, weemploy the method of commuting flows introduced in [37]. In stark contrast withour arguments in [37], an essential ingredient in this paper is thedemonstration of a local smoothing effect for both equations. Despite thenon-perturbative nature of the well-posedness, the gain of derivatives matchesthat of the underlying linear equation. To compensate for the local nature ofthe smoothing estimates, we also demonstrate tightness of orbits. The proofs ofboth local smoothing and tightness rely on our discovery of a new one-parameterfamily of coercive microscopic conservation laws that remain meaningful at thislow regularity.

Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb R)$