Bi and trilinear Schroedinger estimates in one space dimension with applications to cubic NLS and DNLS
Abstract
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 wellposedness 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'_\xi}, 1/r + 1/r'=1. Similar arguments, combined with a gauge transform, lead to local wellposedness 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505457
 arXiv:
 arXiv:math/0505457
 Bibcode:
 2005math......5457G
 Keywords:

 Mathematics  Analysis of PDEs;
 35Q55
 EPrint:
 23 pages