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

Abstract

We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$ . Well-posedness has long been known for $s\geq 0$ , see [55], but not previously for any $s<0$ . The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48].

Keywords

• 35Q55
• 35Q53
• 37K10