Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation
| dc.creator | Bejenaru, Ioan | |
| dc.creator | Tao, Terence | |
| dc.date | 2005-08-11 | |
| dc.date | 2007-10-29 | |
| dc.date.accessioned | 2026-07-07T08:38:56Z | |
| dc.date.available | 2026-07-07T08:38:56Z | |
| dc.description | We establish that the quadratic non-linear Schrödinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s > -3/4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate. | |
| dc.description | 28 pages, no figures, to appear, J.Func. Anal. Some minor gaps filled in | |
| dc.identifier | https://arxiv.org/abs/math/0508210 | |
| dc.identifier | http://arxiv.org/abs/math/0508210 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140911 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35J10 | |
| dc.title | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation | |
| dc.type | text |