Almost sure wellposedness of the cubic nonlinear Schrödinger equation below L^2(T)
Abstract
We consider the Cauchy problem for the onedimensional periodic cubic nonlinear Schrödinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local wellposedness of NLS almost surely for the initial data in the support of the canonical Gaussian measures on H^s(T) for each s > 1/3, and global wellposedness for each s > 1/12.
 Publication:

arXiv eprints
 Pub Date:
 April 2009
 arXiv:
 arXiv:0904.2820
 Bibcode:
 2009arXiv0904.2820C
 Keywords:

 Mathematics  Analysis of PDEs;
 35Q55;
 37K05;
 37L50;
 37L40
 EPrint:
 36 pages. Deterministic multilinear estimates are now summarized in Sec. 3. We use X^{s, b} with b = 1/2+ instead of Z^{s, 1/2}. To appear in Duke Math. J