Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
Abstract
We consider the Cauchy problem for the one-dimensional 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 well-posedness 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 well-posedness for each s > -1/12.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2009
- DOI:
- 10.48550/arXiv.0904.2820
- arXiv:
- arXiv:0904.2820
- Bibcode:
- 2009arXiv0904.2820C
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35Q55;
- 37K05;
- 37L50;
- 37L40
- E-Print:
- 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