Bi- and trilinear Schroedinger estimates in one space dimension with applications to cubic NLS and DNLS
| dc.creator | Gruenrock, Axel | |
| dc.date | 2005-05-22 | |
| dc.date.accessioned | 2026-07-07T05:20:08Z | |
| dc.date.available | 2026-07-07T05:20:08Z | |
| dc.description | The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r'_ξ}, 1/r + 1/r'=1. Similar arguments, combined with a gauge transform, lead to local well-posedness of the Cauchy problem for the derivative nonlinear Schroedinger (DNLS) equation with data u_0 \in ^H^r_{1/2}. In the local result on cubic NLS the parameter r is allowed in the whole subcritical range 1<r<\infty, while for DNLS we assume 1<r \le 2. In the special case r=2 both results coincide with the optimal ones on the H^s - scale. Furthermore, concerning the cubic NLS equation, it is shown by a decomposition argument that the local solution extends globally, provided 2 \ge r > 5/3. | |
| dc.description | 23 pages | |
| dc.identifier | https://arxiv.org/abs/math/0505457 | |
| dc.identifier | http://arxiv.org/abs/math/0505457 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75270 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Q55 | |
| dc.title | Bi- and trilinear Schroedinger estimates in one space dimension with applications to cubic NLS and DNLS | |
| dc.type | text |