We prove that the cubic nonlinear Schrodinger equation (both focusing and defocusing) is globally well-posed in $H^s(mathbb R)$ for any regularity $s>-frac12$. Well-posedness has long been known for $sgeq 0$, see [51], but not previously for any $s<0$. The scaling-critical value $s=-frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 38, 46]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(mathbb R)$ for any $s>-frac12$. The best regularity achieved previously was $sgeq tfrac14$; see [15, 24, 32, 38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a one-parameter family of microscopic conservation laws that remain meaningful at this low regularity.
Download