Sharp wellposedness 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 wellposed in $H^s(\mathbb R)$ for any regularity $s>\frac12$. Wellposedness has long been known for $s\geq 0$, see [55], but not previously for any $s<0$. The scalingcritical value $s=\frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) wellposedness of the real and complexvalued modified Kortewegde 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 datatosolution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the wellposedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new oneparameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.05011
 arXiv:
 arXiv:2003.05011
 Bibcode:
 2020arXiv200305011H
 Keywords:

 Mathematics  Analysis of PDEs;
 35Q55;
 35Q53
 EPrint:
 88 pages