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.

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

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